Data modulation schemes based on the zak transform

ABSTRACT

One example wireless communication method includes transforming an information signal to a discrete sequence, where the discrete sequence is a Zak transformed version of the information signal, generating a first ambiguity function corresponding to the discrete sequence, generating a second ambiguity function by pulse shaping the first ambiguity function, generating a waveform corresponding to the second ambiguity function, and transmitting the waveform over a wireless communication channel. Another communication method includes transforming an information signal to a discrete lattice domain signal, shaping bandwidth and duration of the discrete lattice domain signal by a two-dimensional filtering procedure to generate a filtered information signal, generating, using a Zak transform, a time domain signal from the filtered information signal, and transmitting the time domain signal over a wireless communication channel.

CROSS-REFERENCE TO RELATED APPLICATIONS

This patent document is a continuation of PCT Application No.PCT/US2018/041616 entitled “DATA MODULATION SCHEMES BASED ON THE ZAKTRANSFORM” filed on Jul. 11, 2018, which claims priority to and benefitsof U.S. Provisional Patent Application No. 62/531,808 entitled “RADARWAVEFORM DESIGN IN A ZAK REALIZATION” filed on Jul. 12, 2017. The entirecontents of the aforementioned patent applications are incorporated byreference as part of the disclosure of this patent document.

TECHNICAL FIELD

The present document relates to wireless communication, and moreparticularly, to data modulations schemes used in wirelesscommunication.

BACKGROUND

Due to an explosive growth in the number of wireless user devices andthe amount of wireless data that these devices can generate or consume,current wireless communication networks are fast running out ofbandwidth to accommodate such a high growth in data traffic and providehigh quality of service to users.

Various efforts are underway in the telecommunication industry to comeup with next generation of wireless technologies that can keep up withthe demand on performance of wireless devices and networks.

SUMMARY

This document discloses techniques that can be used to implementtransmitters and receivers for communicating using a modulationtechnique called lattice division multiplexing.

In one example aspect, wireless communication method, implementable by awireless communication apparatus is disclosed. The method includestransforming an information signal to a discrete sequence, where thediscrete sequence is a Zak transformed version of the informationsignal, generating a first ambiguity function corresponding to thediscrete sequence, generating a second ambiguity function by pulseshaping the first ambiguity function, generating a waveformcorresponding to the second ambiguity function, and transmitting thewaveform over a wireless communication channel.

In another example aspect, a wireless communication method,implementable by a wireless communication apparatus is disclosed. Themethod includes transforming an information signal to a discrete latticedomain signal, shaping bandwidth and duration of the discrete latticedomain signal by a two-dimensional filtering procedure to generate afiltered information signal, generating, using a Zak transform, a timedomain signal from the filtered information signal, and transmitting thetime domain signal over a wireless communication channel.

In yet another example aspect, a wireless communication apparatus thatimplements the above-described methods is disclosed.

In yet another example aspect, the method may be embodied asprocessor-executable code and may be stored on a computer-readableprogram medium.

These, and other, features are described in this document.

DESCRIPTION OF THE DRAWINGS

Drawings described herein are used to provide a further understandingand constitute a part of this application. Example embodiments andillustrations thereof are used to explain the technology rather thanlimiting its scope.

FIG. 1 shows an example of a wireless communication system.

FIG. 2 pictorially depicts relationship between time, frequency and Zakdomains.

FIG. 3 shows an example of a hexagonal lattice. The hexagon at thecenter region encloses Voronoi region around the zero lattice point. Thetwo lattice points with arrow decoration are the basis of the maximalrectangular sub-lattice.

FIG. 4 pictorially depicts the periodic and quasi-periodic nature of aninformation grid in the Zak domain.

FIG. 5 is a graphical representation of an OTFS waveform.

FIG. 6 is a graphical representation of filtered OTFS waveform.

FIG. 7 is a graphical comparison of transmit waveforms of a single QAMsymbol using OTFS and OTFS-MC (multicarrier).

FIG. 8 pictorially depicts an example in which the Zak domain and thetime/frequency Zak transforms realizing the signal space realizationlying in between time and frequency realizations.

FIG. 9 shows a depiction of the Zak to generalized Zak interwiningtransformation.

FIG. 10 shows an example of a nested system of lattices with N=3 andM=2.

FIG. 11 is a graphical depiction of an example of a chirp latticecorresponding to N=3, M=2, a=1. The dashed square designated clearregion around zero.

FIG. 12 is a depiction of an example of an ambiguity function of acontinuous Zak chirp corresponding to slope α=1.

FIG. 13 is a flowchart of an example of a wireless communication method.

FIG. 14 is a flowchart of another example of a wireless communicationmethod.

FIG. 15 shows an example of a wireless transceiver apparatus.

DETAILED DESCRIPTION

To make the purposes, technical solutions and advantages of thisdisclosure more apparent, various embodiments are described in detailbelow with reference to the drawings. Unless otherwise noted,embodiments and features in embodiments of the present document may becombined with each other.

Section headings are used in the present document to improve readabilityof the description and do not in any way limit the discussion oftechniques or the embodiments to the respective sections only.

Traditional multi-carrier (MC) transmissions schemes such as orthogonalfrequency division multiplexing (OFDM) schemes are characterized by twoparameters: symbol period (or repetition rate) and subcarrier spacing.The symbols include a cyclic prefix (CP), whose size typically dependson the delay of the wireless channel for which the OFDM modulationscheme is being used. In other words, CP size is often fixed based onchannel delay and if symbols are shrunk to increase system rate, itsimply results in the CP becoming a greater and greater overhead.Furthermore, closely placed subcarriers can cause inter-carrierinterference and thus OFDM systems have a practical limit on how closethe subcarriers can be placed to each other without causing unacceptablelevel of interference, which makes it harder for a receiver tosuccessfully receive the transmitted data.

Furthermore, in traditional cellular communication networks, orthogonalcodes are used by transmitting devices when attempting to join awireless network using a random access mechanism. These orthogonal codesare selected to enable unambiguous detection of the transmitting deviceat a receiving base station.

Embodiments of the disclosed technology are motivated, in part, by therealization that wireless devices may attempt to join the network whilethe channel between the wireless device and a base station may beimpaired both in delay and in Doppler domains due to the movement of thewireless device and multi-path echoes between the wireless device andthe base station. In a similar manner, the theoretical framework foroperation of radars in detecting objects that could be moving, alsobenefits from waveforms that show similar robustness properties as therandom access waveforms in the wireless domain. The present patentdocument provides a theoretical basis for generation of waveforms forsuch use, and other uses, based on techniques for selecting a digitaldomain sequence that is filtered to produce an analog waveform and themathematical relationship between the filter response, and the digitaland analog waveforms as is applied to the situation where practicalsystems attempt to overcome delay and Doppler domain distortions.

The theoretical framework disclosed in the present document may be usedto build signal transmission and reception equipment that can overcomethe above discussed problems, among others.

This patent document discloses, among other techniques, a latticedivision multiplexing technique that, in some embodiments, can be usedto implement embodiments that can perform multi-carrier digitalcommunication without having to rely on CP.

For the sake of illustration, many embodiments disclosed herein aredescribed with reference to the Zak transform. However, one of skill inthe art will understand that other transforms with similar mathematicalproperties may also be used by implementations. For example, suchtransforms may include transforms that can be represented as an infiniteseries in which each term is a product of a dilation of a translation byan integer of the function and an exponential function.

FIG. 1 shows an example communication network 100 in which the disclosedtechnologies can be implemented. The network 100 may include a basestation transmitter that transmits wireless signals s(t) (e.g., downlinksignals) to one or more receivers 102, the received signal being denotedas r(t), which may be located in a variety of locations, includinginside or outside a building and in a moving vehicle. The receivers maytransmit uplink transmissions to the base station, typically locatednear the wireless transmitter. The technology described herein may beimplemented at a receiver 102 or at the transmitter (e.g., a basestation).

Signal transmissions in a wireless network may be represented bydescribing the waveforms in the time domain, in the frequency domain, orin the delay-Doppler domain (e.g., Zak domain). Because these threerepresent three different ways of describing the signals, signal in onedomain can be converted into signal in the other domain via a transform.For example, a time-Zak transform may be used to convert from Zak domainto time domain. For example, a frequency-Zak transform may be used toconvert from the Zak domain to the frequency domain. For example, theFourier transform (or its inverse) may be used to convert between thetime and frequency domains.

The sections designated “A”, “B” and “C” below provide additionalmathematical properties and practical uses of the signal waveforms andgraphs depicted in FIGS. 2 to 12.

The use of the cyclic prefix (CP) has enabled multicarrier waveforms(e.g., OFDM) to operate in frequency-selective channels (when the CP isgreater than the delay spread of the channel), but as the wirelesschannel gets harsher, the length of the CP needs to increase, therebyadding to the overhead of the waveform. As described in Sections “A” and“B”, embodiments of the disclosed technology may be used to performmulti-carrier digital communication without having to rely on CP, so asto advantageously reduce the overhead required by the waveform.

The performance of radar waveforms is particularly susceptible inchannels that are affected by both a delay spread and a doppler spread(e.g., what is termed a “doubly-spread channel”). As described inSection “C”, embodiments of the disclosed technology provide compressedradar waveforms that exhibit a uniform temporal power profile and athumbtack-like ambiguity function with a clean punctured region aroundthe origin who dimensions are free parameters, thereby providinglocalization in the delay-Doppler representation.

A0. Introduction to OTFS Modulation from Zak Theoretic Point

Next few sections explain the OTFS modulation from the Zak theoreticpoint of view. This line of exposition push to the forefront theindependent status of OTFS as a novel modulation technique and revealsits unique mathematical attributes. This, in contrast to the alternativeapproach of presenting OTFS as a preprocessing step over MC modulationwhich somehow obscures the true nature of OTFS and also sacrifice someof its unique strengths. We focus our attention on the following coretheoretical topics:

(1) Heisenberg theory.

(2) Zak theory.

(3) OTFS modulation.

(4) Symplectic Fourier duality relation between OTFS and Multi Carriermodulations which is a particular case of the general relation betweenRadar theory and communication theory.

Before proceeding into a detailed development, it is beneficial to givea brief outline. In signal processing, it is traditional to representsignals (or waveforms) either in time or in the frequency domain. Eachrepresentation reveals different attributes of the signal. Thedictionary between these two realizations is the Fourier transform:

FT:L ₂(t∈

)→L ₂(f∈

)  (0.1)

Interestingly, there is another domain where signals can be naturallyrealized. This domain is called the delay Doppler domain. For thepurpose of the present discussion, this is also referred to as the Zakdomain. In its simplest form, a Zak signal is a function φ(τ, v) of twovariables. The variable τ is called delay and the variable v is calledDoppler. The function φ(τ, v) is assumed to be periodic along v withperiod and v_(r) quasi-periodic along τ with period τ_(r). The quasiperiodicity condition is given by:

φ(τ+nτ _(r1) v+mv _(r))=exp (j2πnv·τ _(r))φ(τ,v),   (0.2)

for every n,m∈

. The periods are assumed to satisfy the Nyquist conditionτ_(r)·v_(r)=1. Zak domain signals are related to time and frequencydomain signals through canonical transforms

_(t) and

_(f) called the time and frequency Zak transforms. In more preciseterms, denoting the Hilbert space of Zak signals by

_(z), the time and frequency Zak transforms are linear transformations:

_(t):

_(z) →L ₂(t∈

),  (0.3)

_(f):

_(z) →L ₂(f∈

),  (0.4)

The pair and

_(t) and

_(f) establishes a factorization of the Fourier transform FT=

_(t)○[

_(f)]⁻¹. This factorization is sometimes referred to as the Zakfactorization. The Zak factorization embodies the combinatorics of thefast Fourier transform algorithm. The precise formulas for the Zaktransforms will be given in the sequel. At this point it is enough tosay that they are principally geometric projections: the time Zaktransform is integration along the Doppler variable and reciprocally thefrequency Zak transform is integration along the delay variable. Thedifferent signal domains and the transformations connecting between themare depicted in FIG. 2.

We next proceed to give the outline of the OTFS modulation. The keything to note is that the Zak transform plays for OTFS the same role theFourier transform plays for OFDM. More specifically, in OTFS, theinformation bits are encoded on the delay Doppler domain as a Zak signalx(τ,v) and transmitted through the rule:

OTFS(x)=

(w* _(94 x)(τ,v)),  (0.5)

where w*_(σ)x(τ,v) stands for two-dimensional filtering operation with a2D pulse w(τ,v) using an operation *σ called twisted convolution (to beexplained in the present document). The conversion to the physical timedomain is done using the Zak transform. Formula (0.5) should becontrasted with the analogue formulas in case of frequency divisionmultiple access FDMA and time division multiple access TDMA. In FDMA,the information bits are encoded on the frequency domain as a signalx(f) and transmitted through the rule:

FDMA(x)=FT(w(f)*x(f)),  (0.6)

where the filtering is done on the frequency domain by linearconvolution with a 1D pulse w(f) (in case of standard OFDM w(f) is equalan sinc function). The modulation mapping is the Fourier transform. InTDMA, the information bits are encoded on the time domain as a signalx(t) and transmitted through the rule:

TDMA(x)=Id(w(t)*x(t))  (0.7)

where the filtering is done on the time domain by linear convolutionwith a 1D pulse w(t). The modulation mapping in this case is identity.

A1. Heisenberg Theory

In this section we introduce the Heisenberg group and the associatedHeisenberg representation. These constitute the fundamental structuresof signal processing. In a nutshell, signal processing can be cast asthe study of various realizations of signals under Heisenberg operationsof delay and phase modulation.

A1.1 The Delay Doppler plane. The most fundamental structure is thedelay Doppler plane V=

equipped with the standard symplectic form:

ω(v ₁ ,v ₂)=v ₁τ₂−τ₁ v ₂,

for every v₁=(τ₁,v₁) and v₂=(τ₂,v₂). Another way to express ω is toarrange the vectors and v₁ and v₂ as the columns of a 2×2 matrix so thatω(v₁,v₂) is equal the additive inverse of the matrix determinant.

${{\omega \left( {v_{1},v_{2}} \right)} = {- {\det \begin{bmatrix}\vdots & \vdots \\v_{1} & v_{2} \\\vdots & \vdots\end{bmatrix}}}},$

The symplectic form is anti-symmetric ω(v₁,v₂)=−ω(v₂,v₁), thus, inparticular ω(v,v)=0 for every v∈V. We also consider the polarizationform:

β(v ₁ ,v ₂)=v ₁τ₂,  (1.2)

for every v₁=(τ₁,v₁) and v₂=(τ₂,v₂). We have that:

β(v ₁ ,v ₂)−β(v ₂ ,v ₁)=ω(v ₁ ,v ₂),

The form β should be thought of as “half” of the symplectic form.Finally, we denote by ψ(

)=exp(2πiz) is the standard one-dimensional Fourier exponent.

A1.2 The Heisenberg group. The polarization form β gives rise to a twostep unipotent group called the Heisenberg group. As a set, theHeisenberg group is realized as Heis=V×S¹ where the multiplication ruleis given by:

(v ₁,

₁)·(v ₂,

₂)=(v ₁ +v ₂,exp(j2πβ(v ₁ ,v ₂))

₁

₂),  (1.4)

One can verify that indeed rule (1.4) yields a group structure: it isassociative, the element (0,1) acts as unit and the inverse of theelement (v,

) is given by:

(v,

)⁻¹=(−v,exp(j2πβ(v,v))

⁻¹)

Most importantly, the Heisenberg group is not commutative. In general,(v₁,

₁) (v₂,

₂)≠(v₂,

₂)·(v₁·

₁) The center consists of all elements of the form (0,

),

∈S¹. The multiplication rule gives rise to a group convolution operationbetween functions:

$\begin{matrix}\begin{matrix}{{h_{1}*_{\sigma}{h_{2}(v)}} = {\int\limits_{{v_{1} + v_{2}} = v}{{{\exp \left( {j\; 2\pi \; {\beta \left( {v_{1},v_{2}} \right)}} \right)}/{h_{1}\left( v_{1} \right)}}{h_{2}\left( v_{2} \right)}}}} \\{{= {\int\limits_{v^{\prime}}{{\exp \left( {j\; 2\pi \; {\beta \left( {v^{\prime},{v - v^{\prime}}} \right)}} \right)}{h_{1}\left( v^{\prime} \right)}{h_{2}\left( {v - v^{\prime}} \right)}d\; v^{\prime}}}},}\end{matrix} & (1.5)\end{matrix}$

for every pair of functions h₁,h₂∈

(V). We refer to (1.5) as Heisenberg convolution or twisted convolution.We note that a twisted convolution differs from linear convolutionthrough the additional phase factor exp (j2πβ,v₁,v₂)).

A1.3 The Heisenberg representation The representation theory of theHeisenberg group is relatively simple. In a nutshell, fixing the actionof the center, there is a unique (up-to isomorphism) irreduciblerepresentation. This uniqueness is referred to as the Stone-von Neumannproperty. The precise statement is summarized in the following theorem:Theorem 1.1 (Stone-von-Neumann Theorem). There is a unique (up toisomorphism) irreducible Unitary representation π:Heis→U(

) such that π(0,

)=

.

In concrete terms, the Heisenberg representation is a collection ofunitary operators π(v)∈U(

), for every v∈V satisfying the multiplicativity relation:

π(v ₁)○π(v ₂)=exp(j2πβ(v ₁ ,v ₂))π(v ₁ +v ₂),  (1.6)

for every v₁,v₂∈V. In other words, the relations between the variousoperators in the family are encoded in the structure of the Heisenberggroup. An equivalent way to view the Heisenberg representation is as alinear transform Π:

(V)→Op(

), taking a function h∈

(V) and sending it to the operator Π(h)∈OP(

) given by:

$\begin{matrix}{{{\prod h} = {\int\limits_{v \in V}{{h(v)}d\; v}}},} & (1.7)\end{matrix}$

The multiplicativity relation (1.6) translates to the fact that Πinterchanges between Heisenberg convolution of functions and compositionof linear transformations, i.e.,

Π(h ₁*_(σ) h ₂)=Π(h ₁)○Π(h ₂),  (1.8)

for every h₁,h₂∈

(V). Interestingly, the representation π, although is unique, admitsmultitude of realizations. Particularly well known are the time andfrequency realizations, both defined on the Hilbert space of complexvalued functions on the real line

=L₂(

). For every x∈

, we define two basic unitary transforms:

L _(x)(φ)(y)=φ(y−x),  (1.9)

M _(x)(φ)(y)=exp(j2πxy)φ(y),  (1.10)

for every φ∈

. The transform L_(x)is called delay by x and the transform M_(x)iscalled modulation by x. Given a point v=(τ,v)∈V we define the timerealization of the Heisenberg representation by:

π_(t)(v)

φ=L _(τ) ○M _(v)(φ),  (1.11)

where we use the notation

to designate the application of an operator on a vector. It is usual inthis context to denote the basic coordinate function by t (time). Underthis convention, the right-hand side of (1.11) takes the explicit formexp (j2πv(t−r))φ(t−γ). Reciprocally, we define the frequency realizationof the Heisenberg representation by:

πf(v)

φ=M _(−τ) ○L _(v)(φ),  (1.12)

In this context, it is accustom to denote the basic coordinate functionby f (frequency). Under this convention, the right-hand side of (1.12)takes the explicit form exp (−j2πτf)φ(f−v). By Theorem 1.1, the time andfrequency realizations are isomorphic in the sense that there is anintertwining transform translating between the time and frequencyHeisenberg actions. The intertwining transform in this case is theFourier transform:

FT(φ)(f)=∫_(t)exp (−j2πft)φ(t)dt,  (1.13)

for every φ∈

. The time and frequency Heisenberg operators π_(t) (v,

) and π_(f)(v,

) are interchanged via the Fourier transform in the sense that:

FT○π_(t)(v)=π_(f)(v)○FT,  (1.14)

for every v∈V. We stress that from the point of view of representationtheory the characteristic property of the Fourier transform is theinterchanging equation (1.14).

A2. Zak Theory

In this section we describe the Zak realization of the signal space. AZak realization depends on a choice of a parameter. This parameter is acritically sampled lattice in the delay Doppler plane. Hence, first wedevote some time to get some familiarity with the basic theory oflattices. For simplicity, we focus our attention on rectangularlattices.

A2.1 Delay Doppler Lattices

A delay Doppler lattice is an integral span of a pair of linearindependent vectors g₁,g₂∈V. In more details, given such a pair, theassociated lattice is the set:

Λ={a ₁ g ₁ +a ₂ g ₂ :a ₁ ,a ₂∈

},  (2.1)

The vectors g₁ and g₂ are called the lattice basis vectors. It isconvenient to arrange the basis vectors as the first and second columnsof a matrix G, i.e.,:

$\begin{matrix}{{G = \begin{bmatrix}\vdots & \vdots \\g_{1} & g_{2} \\\vdots & \vdots\end{bmatrix}},} & (2.2)\end{matrix}$

referred to as the basis matrix. In this way the lattice Λ=G(

²), that is, the image of the standard lattice under the matrix G. Thevolume of the lattice is by definition the area of the fundamentaldomain which is equal to the absolute value of the determinant of G.Every lattice admits a symplectic reciprocal lattice, aka orthogonalcomplement lattice that we denote by Λ¹⁹⁵ . The definition of Λ^(⊥) is:

Λ^(⊥) ={v∈V:ω(v,λ)∈

for every λ∈Λ}  (2.3)

We say that Λ is under-sampled if Λ⊂Λ^(⊥). we say that Λ is criticallysampled if Λ=Λ^(⊥). Alternatively, an under-sampled lattice is such thatthe volume of its fundamental domain is ≥1. From this point on weconsider only under-sampled lattices. Given a lattice Λ, we define itsmaximal rectangular sub-lattice as Λ_(r) =

τ_(r)⊕

v_(r) where:

τ_(r)=arg min{τ>0:(τ,0)∈Λ},  (2.4)

v _(r)=arg min{v>0:(0,v)∈Λ},  (2.5)

When either τ_(r) or v_(r), are infinite, we define Λ_(r)={0}. We say alattice Λ is rectangular if Λ=Λ_(r). Evidently, a sub-lattice of arectangular lattice is also rectangular. A rectangular lattice isunder-sampled if τ_(r)v_(r)≥1. The standard example of a criticallysampled rectangular lattice is Λ_(rec)=

⊕

, generated by the unit matrix:

$\begin{matrix}{{G_{rec} = \begin{bmatrix}1 & 0 \\0 & 1\end{bmatrix}},} & (2.6)\end{matrix}$

An important example of critically sampled lattice that is notrectangular is the hexagonal lattice Λ_(hex), see FIG. 3, generated bythe basis matrix:

$\begin{matrix}{{G_{hex}\begin{bmatrix}a & {a\text{/}2} \\0 & a^{- 1}\end{bmatrix}},} & (2.7)\end{matrix}$

where

$a = \sqrt{2/\sqrt{3}}$

The interesting attribute of the hexagonal lattice is that among allcritically sampled lattices it has the longest distance betweenneighboring points. The maximal rectangular sub-lattice of Λ_(hex) isgenerated by g₁ and 2g₂−g₁, see the two lattice points decorated witharrow heads in FIG. 3. From this point on we consider only rectangularlattices.

A2.2 Zak waveforms A Zak realization is parametrized by a choice of acritically sampled lattice:

Λ=

(τ_(r),0)⊕

(0,v _(r)),  (2.8)

where τ_(r)·v_(r)=1. The signals in a Zak realization are called Zaksignals. Fixing the lattice Λ, a Zak signal is a function φ:V→

that satisfies the following quasi periodicity condition:

φ(v+λ)=exp (j2πβ(v,λ))φ(v),  (2.9)

for every v∈V and λ∈Λ, Writing λ=(kτ_(r),lv_(r)), condition (2.9) takesthe concrete form:

φ(τ+kτ _(r) ,v+lv _(r))=exp(j2πvkτ _(r))φ(τ,v),  (2.10)

that is to say that φ is periodic function along the Doppler dimensionwith period V_(r) and quasi-periodic function along the delay dimensionwith quasi period τ_(r). In conclusion, we denote the Hilbert space ofZak signals by

_(z).

A2.3 Heisenberg action The Hilbert space of Zak signals supports arealization of the Heisenberg representation. Given an element u∈V, thecorresponding Heisenberg operator

(u) is given by:

{π_(z)(u)

φ}(v)=exp(j2πβ(u,v−u))φ(v−u),  (2.11)

for every φ∈

. In words, the element u acts through two-dimensional shift incombination with modulation by a linear phase. The Heisenberg actionsimplifies in case the element u belongs to the lattice. A directcomputation reveals that in this case the action of u=λ∈Λ takes theform:

{π_(z)(λ)

φ}(v)=exp (j2πω(λ,v))φ(v)  (2.12)

In words, the operator π_(z)(λ) is multiplication with the symplecticFourier exponent associated with the point λ. Consequently, the extendedaction of an impulse function h∈

(V) is given by:

$\begin{matrix}\begin{matrix}{{\left\{ {{\prod_{z}(h)} \vartriangleright \phi} \right\} (v)} = {\int\limits_{u \in V}{{h(u)}\left\{ {{\pi_{z}(u)} \vartriangleright \phi} \right\} (v)d\; u}}} \\{{= {\int\limits_{u \in V}{{\psi \left( {j\; 2\; \pi \; \beta \; \left( {u,{v - u}} \right)} \right)}{h(u)}{\phi \left( {v - u} \right)}d\; u}}},}\end{matrix} & (2.13)\end{matrix}$

for every φ∈

_(z). In fact,

(h)

φ=h*_(σ)φ, that is to say that the extended action is given by twistedconvolution of the impulse h with the waveform φ.

A2.4 Zak transforms There are canonical intertwining transformsconverting between Zak signals and time/frequency signals, referred toin the literature as the time/frequency Zak transforms. We denote themby:

_(t):

₂ →L ₂(t∈

)  (2.14)

_(f):

_(z) →L ₂(f∈

)  (2.15)

As it turns out, the time/frequency Zak transforms are basicallygeometric projections along the reciprocal dimensions, see FIG. 4. Theformulas of the transforms are as follows:

_(t)(φ)(t)=∫₀ ^(v) ^(r) φ(t,v)dv,  (2.16)

_(f)(φ)(ƒ)=∫₀ ^(τ) ^(r) exp(−j2πƒτ)φ(τ,ƒ)dτ,   (2.17)

for every φ∈

_(z). In words, the time Zak transform is integration along the Dopplerdimension (taking the DC component) for every point of time.Reciprocally, the frequency Zak transform is Fourier transform along thedelay dimension. The formulas of the inverse transforms are as follows:

$\begin{matrix}{{{_{t}^{- 1}(\phi)}\left( {\tau,v} \right)} = {\sum\limits_{n \in {\mathbb{Z}}}{{\exp \left( {{- j}\; 2\; \pi \; v\; \tau_{r}n} \right)}{\phi \left( {\tau + {n\; \tau_{r}}} \right)}}}} & (2.18) \\{{{{_{f}^{- 1}(\phi)}\left( {\tau,v} \right)} = {\sum\limits_{n \in {\mathbb{Z}}}{{\exp \left( {j\; 2\; \pi \; {\tau \left( {{v_{r}n} + v} \right)}} \right)}{\phi \left( {v + {n\; v_{r}}} \right)}}}},} & (2.19)\end{matrix}$

for every φ∈L₂(

). From this point on we will focus only on the time Zak transform andwe will denote it by

=

_(t). As an intertwining transform

interchanges between the two Heisenberg operators π_(z)(v,

) and π_(t)(v,

), i.e.,:

○

(v)=π_(t)(v)○

  (2.20)

for every v∈V. From the point of view of representation theory thecharacteristic property of the Zak transform is the interchangingequation (2.20).

A2.5 Standard Zak signal Our goal is to describe the Zak representationof the

window function:

$\begin{matrix}{{p(t)} = \left\{ {\begin{matrix}1 & {0 \leq t < \tau_{r}} \\0 & {otherwise}\end{matrix},} \right.} & (2.21)\end{matrix}$

This function is typically used as the generator waveform inmulti-carrier modulations (without CP). A direct application of formula(2.18) reveals that P=

⁻¹(p) is given by:

$\begin{matrix}{{{P\left( {\tau,v} \right)} = {\sum\limits_{n \in {\mathbb{Z}}}\; {{\psi\left( {v\; n\; \tau_{r}}\; \right)}{p\left( {\tau - {n\; \tau_{r}}} \right)}}}}\;,} & (2.22)\end{matrix}$

One can show that P(aτ_(r),bv_(r))=1 for every a,b,∈[0,1), which meansthat it is of constant modulo 1 with phase given by a regular stepfunction along τ with constant step given by the Doppler coordinate v.Note the discontinuity of P as it jumps in phase at every integer pointalong delay. This phase discontinuity is the Zak domain manifestation ofthe discontinuity of the rectangular window p at the boundaries.

A3. OTFS

The OTFS transceiver structure depends on the choice of the followingparameters: a critically sampled lattice Λ=

(τ_(r),0)⊕

(0,v_(r)), a filter function w∈

(V) and an information grid specified by N,M∈

. We assume that the filter function factorizes asw(τ,v)=w_(τ)(τ)w_(v)(v) where the delay and Doppler factors are squareroot Nyquist with respect to Δτ=τ_(r)/N and Δv=v_(r)/M respectively. Weencode the information bits as a periodic 2D sequence of QAM symbolsx=x[nΔτ,mΔv] with periods (N,M). Multiplying x by the standard Zaksignal P we obtain a Zak signal xP. A concrete way to think of xP is asthe unique quasi periodic extension of the finite sequence x[nΔτ,mΔv]where n=0, . . . , N−1 and m=0, . . . , M−1. We define the modulatedtransmit waveform as:

$\begin{matrix}\begin{matrix}{{\mathcal{M}(x)} = {\left( {\prod_{z}{(w)*_{\sigma}{x \cdot P}}} \right)}} \\{{= {\left( {w*_{\sigma}{x \cdot P}} \right)}},}\end{matrix} & (3.1)\end{matrix}$

To summarize: the modulation rule proceeds in three steps. In the firststep the information block x is quasi-periodized thus transformed into adiscrete Zak signal. In the second step, the bandwidth and duration ofthe signal are shaped through a 2D filtering procedure defined bytwisted convolution with the pulse w. In the third step, the filteredsignal is transformed to the time domain through application of the Zaktransform. To better understand the structure of the transmit waveformwe apply few simple algebraic manipulations to (3.1). First, we notethat, being an intertwiner (Formula (2.20)), the Zak transform obeys therelation:

(Π_(z)(w)*_(σ) x·P)=Π_(t)(w)

(x·P),  (3.2)

Second, we note that the factorization w(τ,v)=w_(γ)(τ)w_(v)(v) can beexpressed as twisted convolution w=w_(γ)*_(σ)w_(v). Hence, we can write:

$\begin{matrix}\begin{matrix}{{{\prod_{t}(w)} \vartriangleright {\left( {x \cdot P} \right)}} = {{\prod_{t}\left( {w_{\tau}*_{\sigma}w_{v}} \right)} \vartriangleright {\left( {x \cdot P} \right)}}} \\{= {{\prod_{t}\left( w_{\tau} \right)} \vartriangleright \left\{ {{\prod_{t}\left( w_{v} \right)} \vartriangleright {\left( {x \cdot P} \right)}} \right\}}} \\{{= {w_{\tau}*\left\{ {W_{t} \cdot {\left( {x \cdot P} \right)}} \right\}}},}\end{matrix} & (3.3)\end{matrix}$

where W_(t)=FT⁻¹ (w_(v)) and * stands for linear convolution in time. Werefer to the waveform

(x·P) as the bare OTFS waveform. We see from Formula (3.3) that thetransmit waveform is obtained from the bare waveform through windowingin time followed by convolution with a pulse. This cascade of operationsis the time representation of 2D filtering in the Zak domain. It isbeneficial to study the structure of the bare OTFS waveform in the casex is supported on a single grid point (aka consists of a single QAMsymbol), i.e., x=δ(nΔτ,mΔv). In this case, one can show that the barewaveform takes the form:

$\begin{matrix}{{{\left( {x \cdot P} \right)} = {\sum\limits_{K}\; {{\exp \left( {j\; 2\; \pi \; {m\left( {K + {n\text{/}N}} \right)}\text{/}M} \right)}{\delta \left( {{K\; \tau_{r}} + {n\; \Delta \; \tau}} \right)}}}},} & (3.4)\end{matrix}$

In words, the bare waveform is a shifted and phase modulated infinitedelta pulse train of pulse rate v_(r)=τ_(r) ⁻¹where the shift isdetermined by the delay parameter n and the modulation is determined bythe Doppler parameter m. Bare and filtered OTFS waveforms correspondingto a single QAM symbol are depicted in FIG. 5 and FIG. 6 respectively.We next proceed to describe the de-modulation mapping. Given a receivedwaveform φ_(rx), its de-modulated image y=

(φ_(rx)) is defined through the rule:

(φ_(rx))=w ^(★)*_(σ)

⁻¹(φ_(rx)),

where w^(★) is the matched filter given byw^(★)(v)=exp(−j2πβ(v,v))w(−v). We often incorporate an additional stepof sampling y at (nΔτ,mΔv) for n=0, . . . , N−1 and m=0, . . . , M−1.

A3.1 OTFS channel model The OTFS channel model is the explicit relationbetween the input variable x and the output variable y in the presenceof a channel H. We assume the channel transformation is defined asH=Π_(t)(h) where h=h(τ,v) is the delay Doppler impulse response. Thismeans that given a transmit waveform φ_(tx), the received waveformφ_(rx)=H(φ_(tx)) is given by:

$\begin{matrix}{{{\phi_{xx}\; (t\;)} = {\int\limits_{\tau,v}{{h\left( {\tau,v} \right)}{\exp \left( {j\; 2\; \pi \; {v\left( {t - \tau} \right)}} \right)}{\phi_{t\; x}(t)}d\; {\tau d}\; v}}},} & (3.6)\end{matrix}$

If we take the transmit waveform to be φ_(tx)=

(x) then direct computation reveals that:

$\begin{matrix}\begin{matrix}{y = {\; \circ \; H\; \circ {\mathcal{M}(x)}}} \\{= {w\; \bigstar *_{\sigma}{^{- 1}\left( {{\prod_{t}(h)} \vartriangleright {\left( {w*_{\sigma}{x \cdot P}} \right)}} \right)}}} \\{= {w\; \bigstar *_{\sigma}h*_{\sigma}^{- 1}o\; {\left( {w*_{\sigma}{x \cdot P}} \right)}}} \\{{= {w\; \bigstar *_{\sigma}h*_{\sigma}w*_{\sigma}{x \cdot P}}},}\end{matrix} & (3.7)\end{matrix}$

If we denote h_(w)=w^(★)*_(σ)h*_(σ)w then we can write the input-outputrelation in the form:

y=h _(w) * _(σx·P,)  (3.8)

The delay Doppler impulse h_(w) represents the filtered channel thatinteracts with the QAM symbols when those are modulated and de-modulatedthrough the OTFS transceiver cycle. One can show that under some mildassumptions h_(w)is well approximated by h*w⁽²⁾ where * stands forlinear convolution and w²)=w⁵⁶¹ *w is the linear auto-correlationfunction. In case the channel is trivial, that is h=δ(0,0), we get thath_(w=w) ^(★)*_(σ)w˜w⁽²⁾, thus after sampling we get (an approximate)perfect reconstruction relation:

y[nΔτ,mΔv]˜x[nΔτ,mΔv],  (3.9)

for every n=0, . . . , N−1 and m=0, . . . , M−1.

A4. Symplectic Fourier Duality

In this section we describe a variant of the OTFS modulation that can beexpressed by means of symplectic Fourier duality as a pre-processingstep over critically sampled MC modulation. We refer to this variant asOTFS-MC. For the sake of concreteness, we develop explicit formulas onlyfor the case of OFDM without a CR

A4.1 Symplectic Fourier transformWe denote by L₂(V) the Hilbert space ofsquare integrable functions on the vector space V. For every v∈V wedefine the symplectic exponential (wave function) parametrized by v asthe function ψ_(v):V→

given by:

ψ_(v)(u)=exp(j2πw(v,u)),  (4.1)

for every u∈V, Concretely, if v=(τ,v) and u=(τ′,v′) then ψ_(v)(u)=exp(j2π(vτ′−τv′)). Using symplectic exponents we define the symplecticFourier transform as the unitary transformation SF:L₂(V)→L₂(V) given by:

$\begin{matrix}\begin{matrix}{{{{SF}(g)}(\upsilon)} = {\int\limits_{\upsilon^{\prime}}{\overset{\_}{\psi_{\upsilon}\left( \upsilon^{\prime} \right)}{g\left( \upsilon^{\prime} \right)}d\; \upsilon^{\prime}}}} \\{{= {\int\limits_{\upsilon^{\prime}}{{\exp \left( {{- j}\; 2\; {{\pi\omega}\left( {\upsilon,\upsilon^{\prime}} \right)}} \right)}{g\left( \upsilon^{\prime} \right)}d\; \upsilon^{\prime}}}},}\end{matrix} & (4.2)\end{matrix}$

The symplectic Fourier transform satisfies various interestingproperties (much in analogy with the standard Euclidean Fouriertransform). The symplectic Fourier transform converts between linearconvolution and multiplication of functions, that is:

SF(g _(i) *g ₂)=SF(g ₁)·SF (g ₂)  (4.3)

for every g₁, g₂∈L₂(V). Given a lattice ƒ⊂V, the symplectic Fouriertransform maps sampled functions on Λ to periodic function with respectto the symplectic reciprocal lattice Λ^(⊥). That is, if g is sampled andG=SF(g) then G(v+λ^(⊥))=G(v) for every v∈V and λ^(⊥)∈Λ^(⊥). Thisrelation takes a simpler form in case Λ is critically sampled sinceΛ^(⊥)=Λ. Finally, unlike its Euclidean counterpart, the symplecticFourier transform is equal to its inverse, that is SF=SF⁻¹.

A4.2 OTFS-MC. The main point of departure is the definition of thefiltering pulse w and the way it applies to the QAM symbols. To definethe MC filtering pulse we consider sampled window function W:Λ→

on the lattice Λ=

τ_(r)⊕

v_(r). We define w to be the symplectic Fourier dual to W:

w=SF(W),  (4.4)

By definition, w is a periodic function on V satisfying w w(v+λ)=w(v)for every v∈V and λ∈Λ. Typically, W is taken to be a square window with0/1 values spanning over a certain bandwidth B=M·v_(r) and durationT=N·γ_(r). In such a case, w will turn to be a Dirichlet sinc functionthat is Nyquist with respect to the grid Δ_(N,M)=

Δτ⊕

Δv_(z)

, where:

Δτ=τ_(r) /N,  (4.5)

Δv=v _(r) /M,  (4.6)

More sophisticated windows designs can include tapering along theboundaries and also include pseudo-random scrambling phase values. Asbefore, the bits are encoded as a 2D periodic sequence of QAM symbolsx=x[nΔτ,mΔv] with period (N,M). The transmit waveform is defined throughthe rule:

_(MC)(x)=

((w*x)·P),  (4.7)

In words, the OTFS-MC modulation proceeds in three steps. First step,the periodic sequence is filtered by means of periodic convolution withthe periodic pulse w. Second step, the filtered function is converted toa Zak signal by multiplication with the Zak signal P. Third step, theZak signal is converted into the physical time domain by means of theZak transform. We stress the differences from Formula (3.1) where thesequence is first multiplied by P and then filtered by twistedconvolution with a non-periodic pulse. The point is that unlike (3.1),Formula (4.7) is related through symplectic Fourier duality to MCmodulation. To see this, we first note that w*x=SF(W·X) where X=SF(x).This means that we can write:

$\begin{matrix}\begin{matrix}{{\left( {w*x} \right) \cdot P} = {\sum\limits_{\lambda \in \Lambda}\; {{W(\lambda)}{X(\lambda)}{\psi_{\lambda} \cdot P}}}} \\{{= {{\sum\limits_{\lambda \in \Lambda}\; {{W(\lambda)}{X(\lambda)}{\pi_{z}(\lambda)}}} \vartriangleright P}},}\end{matrix} & (4.8)\end{matrix}$

where the first equality is by definition of the Symplectic Fouriertransform and the second equality is by Formula (2.12). We denoteX_(W)=W·X. Having established this relation we can develop (4.7) intothe form:

$\begin{matrix}\begin{matrix}{{\mathcal{M}_{MC}(x)} = {\left( {{\sum\limits_{\lambda \in \Lambda}{{X_{W}(\lambda)}{\pi_{z}(\lambda)}}} \vartriangleright P} \right)}} \\{= {\sum\limits_{\lambda \in \Lambda}{{X_{W}(\lambda)}{\left( {{\pi_{z}(\lambda)} \vartriangleright P} \right)}}}} \\{= {{\sum\limits_{\lambda \in \Lambda}{{X_{W}(\lambda)}{\pi_{t}(\lambda)}}} \vartriangleright {\; (P)}}} \\{{= {{\sum\limits_{\lambda \in \Lambda}{{X_{W}(\lambda)}{\pi_{t}(\lambda)}}} \vartriangleright p}},}\end{matrix} & (4.9)\end{matrix}$

where the third equality is the intertwining property of the Zaktransform and the forth equality is by definition p=

(P). In case of OFDM without CP, the pulse p is given by the squarewindow along the interval [0,γ_(r) ]. Consequently, the last expressionin (4.9) can be written explicitly as:

$\begin{matrix}\begin{matrix}{{\mathcal{M}_{MC}(x)} = {{\sum\limits_{k,l}{{X_{W}\left( {{k\; \tau_{r}},{lv}_{r}} \right)}{\pi_{t}\left( {{k\; \tau_{r}},{lv}_{r}} \right)}}} \vartriangleright 1_{\lbrack{0,\tau_{r}}\rbrack}}} \\{= {{\sum\limits_{k,l}{{X_{W}\left( {{k\; \tau_{r}},{lv}_{r}} \right)}L_{k\; \tau_{r}}M_{{lv}_{r}}}} \vartriangleright 1_{\lbrack{0,\tau_{r}}\rbrack}}} \\{{= {\sum\limits_{k,l}{{X_{W}\left( {{k\; \tau_{r}},{lv}_{r}} \right)}{\exp \left( {j\; 2\pi \; {{lv}_{r}\left( {t - {k\; \tau_{r}}} \right)}} \right)}1_{\lbrack{{k\; \tau_{r}},{{({k + 1})}\tau_{r}}}\rbrack}}}},}\end{matrix} & (4.10)\end{matrix}$

The last expression of (4.10) can be recognized as MC modulation of the(windowed) sequence of Fourier coefficients X_(W). It is interesting tocompare the transmit waveforms of OTFS and OTFS-MC corresponding tosingle QAM symbols. The two structures are depicted in FIG. 7. The mainstructural difference is the presence of discontinuities at the gridpoints

τ_(γ) in the case of OTFS-MC.

B0. Introduction to OTFS Transceiver Operations from Realization TheoryPerspective

In the subsequent sections, we introduce yet another mathematicalinterpretation of the OTFS transceiver from the point of view ofrealization theory. In a nutshell, in this approach one considers thesignal space of waveforms as a representation space of the Heisenberggroup or equivalently as a Hilbert space equipped with collection ofHeisenberg operators, each associated with a different point in thedelay Doppler plane. This representation space admit multitude ofrealizations. The two standard ones are the time and frequencyrealizations and they are related through the one-dimensional Fouriertransform. In communication theory the TDMA transceiver structure isnaturally adapted to the time realization as QAM symbols are multiplexedalong the time coordinate while the OFDM transceiver structure isnaturally adapted to the frequency realization as the QAM symbols aremultiplexed along the frequency coordinate. The main observation isthat, there is a canonical realization lying in between the time andfrequency realizations, called the Zak realization. Interestingly,waveforms in Zak realization are represented as functions on atwo-dimensional delay Doppler domain satisfying certainquasi-periodicity condition. The main message of this note is that theZak realization is naturally adapted to the OTFS transceiver. Viewingthe OTFS transceiver from this perspective extenuates its novel andindependent standing among the other existing transceiver structures.For convenience, we summarize in the following table the main formulaspresented in this note:

(0.1) QP φ (υ + λ) = ψ (β(υ, λ)) π_(e) (λ)⁻¹  

  φ (υ) Z-Heis π^(e) (υ₀)  

  φ (υ) = ψ (−β (υ₀, υ₀)) ψ (β (υ₀, υ)) φ (υ − υ₀) Z-Heis (lattice)π^(e) (λ, ϵ (λ))  

  φ (υ) = ψ (ω (λ,υ)) φ (υ) Zak to time  

 _(time.) 

  (φ) (t) = ∫₀ ^(υ) ^(r) φ (t, υ) dυ time to Zak  

 _(e.time) (φ) (τ, υ) = Σ_(n) ψ (−υτ 

 n) ƒ (τ + nτ_(r)) Zak to freq  

 _(freq,) 

  (φ) (ƒ) = ∫₀ ^(τ) ^(r) φ(−ƒτ) φ (τ, ƒ) dτ freq to Zak  

 _(ϵ, freq) (φ) (τ, υ) = ψ (τυ) Σ_(n) ψ (τυ_(r)n) φ (ƒ + nυ_(r)) N-Zakto Zak  

 _(ϵ, ϵ′) (φ) (τ, υ) = φ₀ (τ, υ) Zak to N-Zak  

 _(ϵ′, ϵ) (φ)₂ (τ, υ) = φ (−υ · i/N) φ (τ + i/N, υ) Z-std window P_(std)(τ, υ) = Σ_(n) ψ (υτ_(r)n) 1_((n,n + 1)) (τ/τ_(r))

indicates data missing or illegible when filed

where the Q abbreviate Quasi and Z abbreviate Zak.

B1. Mathematical Preliminaries

B1.1 The Delay Doppler plane Let V=

² be the delay Doppler plane equipped with the standard symplecticbilinear form ω:V×V→

given by:

ω(v ₁ v ₂)=v ₁τ₂ −τv ₂,  (1.1)

for every v₁=(τ₁,v₁) and v₂=(τ₂,v₂) Another way to express ω is toarrange the vectors v₁ and v₂ as the columns of a 2×2 matrix. Thesymplectic pairing ⋅(v₁,v₂) is equal the additive inverse of thedeterminant of this matrix, i.e.,:

${\omega \left( {v_{1},v_{2}} \right)} = {- {{\det \begin{bmatrix}\vdots & \vdots \\v_{1} & v_{2} \\\vdots & \vdots\end{bmatrix}}.}}$

We note that the symplectic form is anti-symmetric, i.e.,ω(v₁,v₂)=−(v₂,v₁) thus, in particular ω(v,v)=0 for every v∈V. Inaddition, we consider the polarization form β:V×V→

given by:

β(v ₁ ,v ₂)=v ₁τ₂,  (1.2)

for every v₁=(τ₁,v₁) and v₂=(τ₂,v₂). We have that:

β(v _(z) ,v ₂)−β(v ₂ v ₁)=ω(v ₁ ,v ₂)  (1.3)

The form β should be thought of as “half” of the symplectic form.Finally, we denote by ψ(

)=exp(2πiz) is the standard one-dimensional Fourier exponent.

B1.2 Delay Doppler Lattices Refer to Section A2.1 above.

B1.3 The Heisenberg group The polarization form β:V×V→

gives rise to a two-step unipotent group called the Heisenberg group. Asa set, the Heisenberg group is realized as Heis=V×S¹ where themultiplication rule is given by:

(v ₁ ,z ₁)·(v ₂,

₂)=v ₁ +v ₂,ψ(β(v ₁ ,v ₂))

₁

₂),  (1.11)

One can verify that indeed the rule (1.11) induces a group structure,i.e., it is associative, the element (0,1) acts as unit and the inverseof (v,

) (−v,ψ(β(v,v))

⁻¹). We note that the Heisenberg group is not commutative, i.e., {tildeover (v)}₁·{tilde over (v)}₂ is not necessarily equal to {tilde over(v)}₂·{tilde over (v)}₁. center of the group consists of all elements ofthe form (0,

),

∈S¹. The multiplication rule gives rise to a group convolution operationbetween functions:

$\begin{matrix}{{{f_{1}{f_{2}\left( \overset{\sim}{v} \right)}} = {\int_{{v_{1} \cdot v_{2}} = v}{{f_{1}\left( {\overset{\sim}{v}}_{1} \right)}{f_{2\;}\left( {\overset{\sim}{v}}_{2} \right)}}}},} & (1.12)\end{matrix}$

for every pair of functions ƒ₁ƒ₂∈

(Heis). We refer to the convolution operation

as Heisenberg convolution or twisted convolution.

The Heisenberg group admits multitude of finite subquotient groups. Eachsuch group is associated with a choice of a pair (Λ,ϵ) where Λ⊂V is anunder-sampled lattice and ϵ:Λ→S¹ is a map satisfying the followingcondition:

ϵ(λ₁+λ₂)=ϵ(λ₁)ϵ(λ₂)ψ(β(λ₁,λ₂))  (1.13)

Using ϵ we define a section map {circumflex over (ϵ)}:Λ→Heis given by{circumflex over (ϵ)}(λ)=(λ,ϵ(λ)) . One can verify that (1.13) impliesthat {circumflex over (ϵ)} is a group homomorphism, {circumflex over(ϵ)}(λ₁+λ₂)={circumflex over (ϵ)}(λ₁) ·(λ₂). To summarize, the map ϵdefines a sectional homomorphic embedding of Λ as a subgroup of theHeisenberg group. We refer to ϵ as a Heisenberg character and to thepair (Λ,ϵ) as a Heisenberg lattice. A simple example is when the latticeΛ is rectangular, i.e., Λ=Λ_(r). In this situation β|Λ=0 thus we cantake ϵ=1, corresponding to the trivial embedding {circumflex over(ϵ)}(λ)=(λ,1). A more complicated example is the hexagonal latticeΛ=Λ_(hex) equipped with ϵ_(hex):Λ_(hex)→S¹, given by:

ϵ_(hex)(ng ₁ +mg ₂)=ψ(m ²/4),  (1.14)

for every n,m∈

. An Heisenberg lattice defines a commutative subgroup Λ_(ϵ)⊂ Heisconsisting of all elements of the form (Λ,ϵ(λ)), λ∈Λ. The centralizersubgroup of Im{circumflex over (ϵ)} is the subgroup Λ^(⊥)×S¹. We definethe finite subqotient group:

Heis(Λ,ϵ)=Λ^(⊥) ×S ¹ /Im{circumflex over (ϵ)},  (1.15)

The group Heis (Λ,ϵ) is a central extension of the finite commutativegroup Λ^(⊥)/Λ, by the unit circle that S¹, that is, it fits in thefollowing exact sequence:

S ¹→Heis(Λ,ϵ)

Λ^(⊥)/Λ,

We refer to Heis (Λ,ϵ) as the finite Heisenberg group associated withthe Heisenberg lattice (Λ,ϵ). The finite Heisenberg group takes a moreconcrete from in the rectangular case.

Specifically, when Λ=Λ_(r) and ϵ=1, we have Heis(Λ,1)=Λ^(⊥)/Λ×S¹≃

/N×

/N×S¹ with multiplication rule given by:

(k ₁ , l ₁,

₁)·(k ₂ ,l ₂,

₂)=(k ₁ +k ₂ ,l ₁ +l ₂,ψ(l ₁ k ₂ /N)

₁

₂),  (1.17)

B1.4 The Heisenberg representationThe representation theory of theHeisenberg group is relatively simple. In a nutshell, fixing the actionof the center, there is a unique (up-to isomorphism) irreduciblerepresentation. This uniqueness is referred to as the Stone-von Neumannproperty. The precise statement is summarized in Section A1.3:The factthat π is a representation, aka multiplicative, translates to the factthat Π interchanges between group convolution of functions andcomposition of linear transformations, i.e., Π(ƒ₁

ƒ₂)=Π(ƒ₁)○Π(ƒ₂).

Since π(0,

)=

, by Fourier theory, its enough to consider only functions ƒ thatsatisfy the condition ƒ(v,

)=

⁻¹ƒ(v,1). Identifying such functions with their restriction to v=v×{1}we can write the group convolution in the form:

$\begin{matrix}{{{f_{1}{f_{2}(v)}} = {\int_{{v_{1} + v_{2}} = v}{{\psi \left( {\beta \left( {v_{1},v_{2}} \right)} \right)}{f_{1}\left( v_{1} \right)}{f_{2}\left( v_{2} \right)}}}},} & (1.19)\end{matrix}$

Interestingly, the representation π, although is unique, admitsmultitude of realizations. Particularly well known are the time andfrequency realizations which are omnipresent in signal processing. Weconsider the Hilbert space of complex valued functions on the real line

=

(

) . To describe them, we introduce two basic unitary operations on suchfunctions—one called delay and the other modulation, defined as follows:

Delay:L _(x)(φ)(y)=φ(y−x),  (1.20)

Modulation:M _(x)(φ)(y)=ψ(xy)φ(y),  1.21)

for any value of the parameter x∈

and every function φ∈

. Given a point v=(τ,v) we define the time realization of the Heisenbergrepresentation by:

π^(time)(v,

)

φ=

·L _(τ) ○M _(v)(φ)  (1.22)

where we use the notation

to designate the application of an operator on a vector. It is accustomin this context to denote the basic coordinate function by t (time).Under this convention, the right hand side of (1.22) takes the explicitform

ψ(v(t−τ))φ(t−τ). Reciprocally, we define the frequency realization ofthe Heisenberg representation by:

$\begin{matrix}{\mspace{79mu} {{{{\pi^{freq}\left( {\upsilon,\tau} \right)}\mspace{11mu} \mspace{11mu} \phi} = {{z \cdot {{M\_}_{\tau} \circ L}}\text{?}\; (\phi)}},{\text{?}\text{indicates text missing or illegible when filed}}}} & (1.23)\end{matrix}$

In this context, it is accustom to denote the basic coordinate functionby ƒ(frequency). Under this convention, the right hand side of (1.23)takes the explicit form

ψ(−τƒ)φ(ƒ−v). By Theorem 1.1, the time and frequency realizations areisomorphic. The isomorphism is given by the Fourier transform:

FT(φ)(ƒ)=∫_(t)exp(−2πiƒt)φ(t)dt,  (1.24)

for every φ∈

. As an intertwining transform FT interchanges between the twoHeisenberg operators π_(t)(v,

) and π_(f)(v,

), i.e.,:)

FT○π ^(time)(v,

)=π^(freq)(v,

○FT,  (1.25)

for every (v,

). From the point of view of representation theory the characteristicproperty of the Fourier transform is the interchanging equation (1.25).Finally, we note that from communication theory perspective, the timedomain realization is adapted to modulation techniques where QAM symbolsare arranged along a regular lattice of the time domain. Reciprocally,the frequency realization is adapted to modulation techniques (lineOFDM) where QAM symbols are arranged along a regular lattice on thefrequency domain. We will see in the sequel that there exists other,more exotic, realizations of the signal space which give rise to afamily of completely new modulation techniques which we call ZDMA.

The finite Heisenberg representation. It is nice to observe that thetheory of the Heisenberg group carry over word for word to the finiteset-up. In particular, given an Heisenberg lattice (Λ,ϵ), the associatedfinite Heisenberg group Heis (Λ,ϵ) admits a unique up to isomorphismirreducible representation after fixing the action of the center. Thisis summarized in the following theorem.

Theorem 1.2 (Finite dimensional Stone-von Neumann theorem). There is aunique (up to isomorphism) irreducible unitary representationπ_(ϵ):Heis(ƒ,ϵ)→U(

) such that π_(ϵ)(0,

)=

Id

. Moreover π_(ϵ) is finite dimensional with dim

=N where N²=#Λ^(⊥)/Λ.

For the sake of simplicity, we focus our attention on the particularcase where Λ=

(τ_(r),0)⊕

(0,v_(r)) is rectangular and ϵ=1 and proceed to describe the finitedimensional counterparts of the time and frequency realizations ofπ_(ϵ). To this end, we consider the finite dimensional Hilbert space

_(N)=

(

/N) of complex valued functions on the ring

/N, aka—the finite line. Vectors in

_(N) can be viewed of as uniformly sampled functions on the unit circle.As in the continuous case, we introduce the operations of (cyclic) delayand modulation:

L _(n)(φ)(m)=φ(m−n),   (1.26)

M _(n)(φ)(m)=ψ(nm/N)φ(m),  (1.27)

for every φ∈

_(N) and n∈

/N noting that the operation m−n is carried in the cyclic ring

/N. Given a point (k/v_(r), l/τ_(r))∈Λ^(⊥)we define the finite timerealization by:

π_(ϵ) ^(time)(k/v _(r) ,l/τ _(r),

)

φ=

·L _(k)○M_(l)(φ)  (1.28)

Denoting the basic coordinate function by n we can write the right handside of (1.28) in the explicit form

ψ(l(n−k)/N)φ(n−k). Reciprocally, we define the finite frequencyrealization by:

π_(ϵ) ^(freq)(k/v _(r) ,l/τ _(r),

)

φ=

·M _(−k)○L_(l)(φ),  (1.29)

Denoting the basic coordinate function by m the right hand side of(1.29) can be written in the explicit form

ψ(−km/N)φ(m−l). By Theorem 1.2, the discrete time and frequencyrealizations are isomorphic and the isomorphism is realized by thefinite Fourier transform:

$\begin{matrix}{{{{{FFT}(\phi)}(m)} = {\sum\limits_{n = 0}^{N - 1}{{\exp \left( {{- 2}\; \pi \; {{imn}/N}} \right)}{\phi (n)}}}},} & (1.30)\end{matrix}$

As an intertwining transform the FFT interchanges between the twoHeisenberg me operators π_(ϵ) ^(time)(v,

) and π_(ϵ) ^(freq)(v,

), i.e.,:

FFT○π _(ϵ) ^(time)(v

)=π_(ϵ) ^(freq)(v,

)○FFT,  (1.31)

for every (v,

)∈=Heis (Λ, ϵ).

B2. The Zak Realization

B2. 1 Zak waveforms See previous discussion in A2.2. In this section wedescribe a family of realizations of the Heisenberg representation thatsimultaneously combine attributes of both time and frequency. These areknown in the literature as Zak type or lattice type realizations. Aparticular Zak realization is parametrized by a choice of an Heisenberglattice (Λ,ϵ) where Λ is critically sampled. A Zak waveform is afunction φ:V→

that satisfies the following quasi periodicity condition:

φ(v+λ)=ϵ(λ)=ϵ(λ)ψ(β(v,λ))φ(v)  (2.1)

There is an alternative formulation of condition (2.1) that is bettersuited when considering generalizations. The basic observation is thatthe map ϵ defines a one dimensional representation π

:Λ×S¹→U(

) satisfying the extra condition that π_(ϵ()0,

)=

. This representation is given by π_(ϵ)(λ,

)=ϵ(λ)⁻¹

. Indeed, verify that:

$\begin{matrix}\begin{matrix}{{{\pi_{\epsilon}\left( {\lambda_{1},z_{1}} \right)} \cdot {\pi_{\epsilon}\left( {\lambda_{2},z_{2}} \right)}} = {{\epsilon \left( \lambda_{1} \right)}^{- 1}{\epsilon \left( \lambda_{2} \right)}^{- 1}z_{1}z_{2}}} \\{= {{\epsilon \left( {\lambda_{1} + \lambda_{2}} \right)}^{- 1}{\psi \left( {\beta \left( {\lambda_{1},\lambda_{2}} \right)} \right)}z_{1}z_{2}}} \\{{= {\pi_{\epsilon}\left( {\left( {\lambda_{1},z_{1}} \right) \cdot \left( {\lambda_{2},z_{2}} \right)} \right)}},}\end{matrix} & (2.2)\end{matrix}$

In addition, we have that π_(ϵ)(λ,ϵ(λ))=1 implying the relation[m{circumflex over (ϵ)}⊂kerπ_(ϵ). Hence π_(ϵ), is in fact arepresentation of the finite Heisenberg group Heis(Λ,ϵ)=Λ×S¹/Im{circumflex over (ϵ)}. Uisng the representation π_(ϵ), wecan express (2.1) in the form:

φ(v+λ)=ψ(β(v,λ)){π_(ϵ(λ)) ⁻¹

(v)},  (2.3)

We denote the Hilbert space of Zak waveforms by

(V,π_(ϵ)) or sometimes for short by 0.15

_(ϵ). For example, in the rectangular situation where Λ=Λ

and ϵ=1, condition (2.1) takes the concrete formφ(τ+kτ_(r),v+lv_(r))=ψ(vkτ_(r))φ(τ,v), that is, φ is periodic functionalong the Doppler dimension (with period v_(r)) and quasi-periodicfunction along the delay dimension. Next, we describe the action of theHeisenberg group on the Hilbert space of Zak waveforms. Given a Zakwaveform φ∈

_(ϵ), and an element (u,

)∈ Heis, the action of the element on the waveform is given by:

{π^(ϵ() u,

)

φ}(v)=

·ψ(β(u,v−u))φ(v−u),  (2.4)

In addition, given a lattice point λ∈Λ, the action of the element{circumflex over (ϵ)}(λ)=(λ, ϵ(λ)) takes the simple form:

$\begin{matrix}\begin{matrix}{{\left\{ {{\pi^{\epsilon}\left( {\lambda,{\epsilon (\lambda)}} \right)} \vartriangleright \phi} \right\} (v)} = {{\epsilon (\lambda)}{\psi \left( {\beta \left( {\lambda,{v - \lambda}} \right)} \right)}{\phi \left( {v - \lambda} \right)}}} \\{= {{\epsilon (\lambda)}{\epsilon \left( {- \lambda} \right)}{\psi \left( {\beta \left( {\lambda,{- \lambda}} \right)} \right)}{\psi \left( {\omega \left( {\lambda,v} \right)} \right)}{\phi (v)}}} \\{{= {{\psi \left( {\omega \left( {\lambda,v} \right)} \right)}{\phi (v)}}},}\end{matrix} & (2.5)\end{matrix}$

where in the first equality we use (2.4), in the second equality we use(2.1) and the polarization equation (1.3) and in the third equality weuse (1.13). To conclude, we see that π^(ϵ)({circumflex over (ϵ)}(λ)) isgiven by multiplication with the symplectic Fourier exponent associatedwith the point λ. As usual, the representation gives rise to an extendedaction by functions on V. Given a function h∈

(V), its action on a Zak waveform

$\begin{matrix}\begin{matrix}{{\left\{ {{\Pi^{\epsilon}(h)} \vartriangleright \phi} \right\} (v)} = {\int_{u \in V}{{h(u)}\left\{ {{\pi^{\epsilon}(u)} \vartriangleright \phi} \right\} (v){du}}}} \\{{= {\int_{u \in V}{{\psi \left( {\beta \left( {u,{v - u}} \right)} \right)}{h(u)}{\phi \left( {v - u} \right)}{du}}}},}\end{matrix} & (2.6)\end{matrix}$

From the last expression we conclude that ∪^(ϵ)

φ=h

φ, namely, the extended action is realized by twisted convolution of theimpulse h with the waveform φ.

B2.2 Zak transforms See also section A2.4. By Theorem 1.1, the Zakrealization is isomorphic both to the time and frequency realizations.Hence there are intertwining transforms interchanging between thecorresponding Heisenberg group actions. These intertwining transformsare usually referred to in the literature as the time/frequency Zaktransforms and we

_(time,ϵ):

_(ϵ)→

_(time)=

(t∈

),  (2.7)

denote them by:

_(freq,ϵ):

_(ϵ)→

_(freq)=

(ƒ∈

),  (2.8)

As it turns out, the time/frequency Zak transforms are basicallygeometric projections along the reciprocal dimensions, see FIG. 2.Formally, this assertion is true only when the maximal rectangularsublattice Λ_(r) =

(r_(r) ,0)⊕

(0,v_(r) ) is non-trivial, i.e., when the rectangular parameters τ_(r),v_(r) <∞. Assuming this condition holds, let N=τ_(r)·v_(r) denote theindex of the rectangular sublattice A_(r) with respect to the fulllattice A, i.e., N=[Λ_(r):Λ]. For example, when Λ=Λ_(rec) we haveτ_(r)=v_(r)=1 and N=1. When Λ=Λ_(hex); we have τ_(r) =a and v_(r) =2/aand consequently N=2. Without loss of generality, we assume that ϵ|_(r)=1.

Granting this assumption, we have the following formulas:

_(time,ϵ)(φ)(t)=∫₀ ^(v) ^(r) φ(t,v)dv,  (2.9)

_(freq,ϵ)(φ)(ƒ)=∫₀ ^(τ) ^(r) ψ(−ƒτ)φ(τ,ƒ)dr,  (2.10)

We now proceed describe the intertwining transforms in the oppositedirection, which we denote by:

_(ϵ,time):

_(time)→

_(ϵ),  (2.11)

_(ϵ,freq):

_(freq)→

_(ϵ),  (2.12)

To describe these we need to introduce some terminology. Let π_(r)^(time) _(r) and b^(freq) denote the time and frequency realizations ofthe Heisenberg representation of the group Heis (A_(r),1)b^(time),b^(freq)∈

^(N) are the unique (up to multiplication by scalar) invariant vectorsunder the action of Λ_(ϵ)through π_(r) ^(time) and π_(r) ^(freq)respectively. The formulas of (2.11) and (2.12) are:

$\begin{matrix}{{{{_{\epsilon,{time}}(\phi)}\left( {\tau,v} \right)} = {\sum\limits_{k = 0}^{N - 1}{\sum\limits_{n \in {\mathbb{Z}}}{{b^{time}\lbrack k\rbrack}{\psi \left( {{- v}\; {\tau_{r}\left( {{k/N} + n} \right)}} \right)}{\phi \left( {\tau + {\tau_{r}\left( {{k/N} + n} \right)}} \right)}}}}},} & (2.13) \\{{{{_{\epsilon,{freq}}(\phi)}\left( {\tau,v} \right)} = {{\psi \left( {\tau \; v} \right)}{\sum\limits_{k = 0}^{N - 1}{\sum\limits_{n \in {\mathbb{Z}}}{{b^{freq}\lbrack k\rbrack}{\psi \left( {\tau \; {v_{r}\left( {{k/N} + n} \right)}} \right)}{\phi \left( {v + {v_{r}\left( {{k/N} + n} \right)}} \right)}}}}}},} & (2.14)\end{matrix}$

In the rectangular situation where Λ=Λ_(r), and ϵ=1, we have N=1 andb^(time)=b^(freq)=1. Substituting these values in (2.13) and (2.14) weget:

$\begin{matrix}{{{{_{\epsilon,{time}}(\phi)}\left( {\tau,v} \right)} = {\sum\limits_{n\; \in {\mathbb{Z}}}{{\psi \left( {{- v}\; \tau_{r}n} \right)}{\phi \left( {\tau + {n\; \tau_{r}}} \right)}}}},} & (2.15) \\{{{{_{\epsilon,{freq}}(\phi)}\left( {\tau,v} \right)} = {{\psi \left( {\tau \; v} \right)}{\sum\limits_{n \in {\mathbb{Z}}}{{\psi \left( {\tau \; v_{r}n} \right)}{\phi \left( {v + {nv}_{r}} \right)}}}}},} & (2.16)\end{matrix}$

In addition, in the hexagonal situation where Λ=Λ_(hex) and ϵ=ϵ_(hex),we have N=2,τ_(r)=a,v_(r)=2a⁻¹ and b^(time)=(1,i). Substituting thesevalues in (2.11) and (2.12) we get:

$\begin{matrix}{{{{_{\epsilon,{time}}(\phi)}\left( {\tau,v} \right)} = {{\sum\limits_{n\; \in {\mathbb{Z}}}{{\psi \left( {- {van}} \right)}{\phi \left( {\tau + {an}} \right)}}} + {i\; {\sum\limits_{n \in {\mathbb{Z}}}{{\psi \left( {- {{va}\left( {{1/2} + n} \right)}} \right)}{\phi \left( {\tau + {a\left( {{1/2} + n} \right)}} \right)}}}}}},} & (2.17) \\{{{_{\epsilon,{freq}}(\phi)}\left( {\tau,v} \right)} = {{{\psi \left( {\tau \; v} \right)}{\sum\limits_{n \in {\mathbb{Z}}}{{\psi \left( {2\; \tau \; a^{- 1}n} \right)}{\phi \left( {v + {2a^{- 1}n}} \right)}}}} - {i\; {\psi \left( {\tau \; v} \right)}{\sum\limits_{n \in {\mathbb{Z}}}{{\psi \left( {2\tau \; {a^{- 1}\left( {n + {1/2}} \right)}} \right)}{\phi \left( {v + {2{a^{- 1}\left( {n + {1/2}} \right)}}} \right)}}}}}} & (2.18)\end{matrix}$

Furthermore, one can show that

_(time,ϵ)○

_(ϵ,freq) ∝FT hence the pair of Zak trans-forms constitute a square rootdecomposition of the Fourier transform, reinforcing the interpretationof the Zak realization as residing between the time and the frequency(see FIG. 8)

As mentioned before, the characteristic property of the Zak transform isthat it interchanges between the Heisenberg group actions:

Proposition 2.1. We have:

_(time,ϵ(π) ^(ϵ)(v,

)

φ)=π^(time)(v,

)

_(time,ϵ(φ),)  (2.19)

_(freq,ϵ)(π^(ϵ)(v,

)

φ)=π^(freq)(v

)

_(freq,ϵ)(φ),   (2.20)

for every φ∈

_(ϵ) and (v,

)∈Heis.

Example 2.2. As an example we consider the rectangular lattice Λ_(r)=

(τ_(r),0)⊕

(0,1/τ_(r)) and the trivial Heisenberg character ϵ=1. Under thesechoices, we describe the Zak realization of the window function:

$\begin{matrix}{{p(t)} = \left\{ {\begin{matrix}1 & {0 \leq t < \tau_{r}} \\0 & {otherwise}\end{matrix},} \right.} & (2.21)\end{matrix}$

This function is typically used as the generator filter in multi-carriermodulations (without CP). A direct application of formula (2.15) revealsthat P=

_(ϵ,time)(p) is given by:

$\begin{matrix}{{{P\left( {\tau,v} \right)} = {\sum\limits_{n \in {\mathbb{Z}}}{{\psi \left( {{vn}\; \tau_{r}} \right)}{p\left( {r - {n\; \tau_{r}}} \right)}}}},} & (2.22)\end{matrix}$

One can show that P(aτ_(r),b/τ_(r))=1 for every a,b∈[0,1), which meansthat it is of constant modulo 1 with phase given by a regular stepfunction along τ with constant step given by the Doppler coordinate v.Note the discontinuity of P as it jumps in phase at every integer pointalong delay. This phase discontinuity is the Zak domain manifestation ofthe discontinuity of the rectangular window p at the boundaries.

B3. The Generalized Zak Realization

For various computational reasons that arise in the context of channelequalization we need to extend the scope and consider also higherdimensional generalizations of the standard scalar Zak realization.Specifically, a generalized Zak realization is a parametrized by anunder-sampled Heisenberg lattice (Λ,ϵ). Given this choice, we fix thefollowing structures:

Let Heis (Λ,ϵ)=Λ^(⊥)×S¹/Λ_(ϵ), be the finite Heisenberg group associatedwith (Λ,ϵ), see Formula (1.15). Let N²=[Λ:Λ^(⊥]) be the index of Λinside Λ^(⊥). Finally, let π_(ϵ) be the finite dimensional Heisenbergrepresentation of Heis (Λ,ϵ). At this point we are not interested in anyspecific realization of the representation π_(ϵ).

A generalized Zak waveform is a vector valued function φ:V→

^(N) that satisfy the following π_(ϵ), quasi-periodicity condition:

φ(v+λ)=ψ(β(v,λ)){π_(ϵ)(λ)⁻¹

φ(v)},  (3.1)

for every v∈V and λ∈Λ^(⊥). Observe that when the lattice A is criticallysampled, we have N=1 and condition (3.1) reduces to (2.3). In therectangular situation where Λ=Λ_(r),ϵ=1 we can take π_(ϵ)=π_(ϵ) ^(time),thus the quasi-periodicity condition (3.1) takes the explicit form:

φ(τ+k/v _(r) ,v+l/γ _(r))=ψ(vk/v _(r)){ψ(kl/N)M _(−l)L_(−k)

φ(τ,v)}  (3.2)

where we substitute v=(τ,v) and λ=(k/v_(r),l/τ_(r)). In particular, wesee from (3.2) that the nth coordinate of φ satisfies the followingcondition along Doppler:

φ_(n)(τ,v+l/r _(r))=ψ(−nl/N)φ_(n)(τ,v),  (3.3)

for every (τ,v)∈V and l∈

. We donate by

_(ϵ)=

(V,π_(ϵ)) the Hilbert space of generalized Zak waveforms. We now proceedto define the action of Heisenberg group on H_(t). The action formula issimilar to (2.4) and is given by:

{π^(ϵ)(v,

)

φ}(v′)=

·ψ(β(v,v′−v)φ(v′−v)  (3.4)

for every φ∈

_(ϵ) and (v,

)∈Heis. Similarly, we have {π^(ϵ)(λ,ϵ(λ))

φ}(v)=ψ(w(λ,v))φ(v), for every λ∈Λ.

3.1 Zak to Zak Intertwining Transforms

The standard and the generalized Zak realizations of the Heisenbergrepresentation are isomorphic in the sense that there exists a non-zerointertwining transform commuting between the corresponding Heisenbergactions. To describe it, we consider the following setup. We fix acritically sampled Heisenberg lattice Λ,ϵ) and a sub-lattice ƒ′⊂Λ ofindex N. We denote by ϵ′the restriction of ϵ to the sub-lattice Λ′. Ourgoal is to describe the intertwining transforms (See FIG. 9):

_(ϵ,ϵ′):

_(ϵ′)→

_(ϵ),  (3.5)

_(ϵ′,ϵ):

_(ϵ)→

_(ϵ′),  (3.6)

We begin with the description of

_(ϵ,ϵ′). Let ζ∈

^(N) be the unique (up-to multiplication by scalar) invariant vectorunder the action of the subgroup V^(ϵ)=ζ(V)⊂ Heis (Λ′,ϵ′) through therepresentation π_(ϵ′), namely, ζ satisfies the condition:

π_(ϵ′)(λ,ϵ(λ))

ζ=ζ,

for every λ∈Λ. Given a generalized Zak waveform φ∈

_(φ′), the transformed waveform

_(ϵ,ϵ′)(φ) is given by:

_(ϵ,ϵ′)(φ)(v)=(ζ,φ(v)),  (3.7)

for every v∈V. In words, the transformed waveform is defined pointwiseby taking the inner product with the invariant vector ζ. We proceed withthe description of

_(ϵ′,ϵ). To this end, we define the Hilbert space of sampled Zakwaveforms. A sampled Zak waveform is a function ϕ:Λ′^(⊥)→

satisfying the following discrete version of the quasi-periodicitycondition (2.3):

ϕ(δ+λ)=ψ(β(δ,λ))π_(ϵ)(λ)⁻¹

ϕ(δ),  (3.8)

for every δ∈Λ′^(⊥) and λ∈Λ. We denote the Hilbert space of sampled Zakwaveforms by

′^(⊥),π_(ϵ)). One can show that

(Λ′^(⊥),π_(ϵ)) is a finite dimensional vector space of dimension[Λ:Λ′^(⊥)]=[Λ′:Λ]=N. The Hilbert space of sampled Zak waveforms admitsan action of the finite Heisenberg group Heis (Λ′,ϵ′). This action is adiscrete version of 2.4) given by:

{π_(ϵ)(δ,

)

ϕ}(δ′)=

ψ(β(δ,δ′−δ))ϕ(δ′−δ),  (3.9)

for every ϕ∈

(Λ′^(⊥),π

), and points δ,δ′∈Λ′^(⊥). We can now define the intertwining transform

_(ϵ′,ϵ).

Given a Zak waveform φ∈

_(ϵ) the transformed generalized waveform φ′=

_(ϵ′,ϵ)(φ) is a function on V taking values in the N dimensional Hilbertspace

(Λ′^(⊥),π_(ϵ))≃

^(N), defined by:

φ′(v)(δ)=ψ(−β(v,δ))φ(v+δ),  (3.10)

for every v∈V and δ∈Λ′^(⊥). For the sake of concreteness, it isbeneficial to describe in detail the rectangular situation. We considera rectangular lattice Λ=Λ_(r), with trivial embedding, ϵ=1 and thesublattice Λ′=

(γ_(r),0)⊕

(0,Nv_(r)). Evidently, we have [Λ′:Λ]=N. For these particular choices,the structures described above take the following concrete form:

-   -   The finite Heisenberg group associated with (Ε,ϵ) is given by:

Heis(Λ,ϵ)=Λ^(⊥) /Λ×S ¹ ≃S ¹,

-   -   The finite Heisenberg representation of Heis (Λ,ϵ), is given by:

π_(ϵ)(

)=

,

-   -   The orthogonal complement lattice of Λ′ is given by:

Λ′^(⊥)=

(τ_(r) /N,0)⊕

(0,v _(r)),

-   -   The finite Heisenbem group associated with (Λ′,ϵ′) is given by:

Heis(Λ′,ϵ′)=Λ′^(⊥) /Λ×S ¹ ≃

/N×

/N×S ¹,

-   -   The finite Heisenberg representation of Heis (Λ′,ϵ′), is given        by π_(ϵ′)=π_(ϵ′) ^(time), where:

π_(ϵ′) ^(time)(kτ _(r) /N,lv _(r),

)=

L _(k) M _(l),

-   -   The invariant vector under π_(ϵ′)(λ))=π_(ϵ′)(λ,ϵ′(λ)),λ∈Λ is        given by:

ζ=δ(0).

Substituting in Formula (3.7), we get:

$\begin{matrix}\begin{matrix}{{{_{\epsilon,\epsilon^{\prime}}(\phi)}(v)} = {\langle{{\phi (0)},{\phi (v)}}\rangle}} \\{= {\phi_{0}(v)}}\end{matrix} & (3.11)\end{matrix}$

In words, the conversion from generalized to standard Zak waveforms is“simply” to take the zero coordinate at each point v∈V. In the oppositedirection, given a Zak waveform φ∈

(V,π_(ϵ)) its restriction to the lattice Λ′^(⊥) is periodic with respectto translations by elements of Λ, hence is a function on the quotientgroup Λ′^(⊥)/Λ=

/N, i.e., a vector in

(

/N) Substituting in Formula (3.10), we get that:

${{{_{\epsilon,\epsilon^{\prime}}(\phi)}\left( {\tau,v} \right)} = \begin{bmatrix}{{\psi \left( {{{- v}\; \tau_{r}},{0/N}} \right)}{\phi \left( {\tau,v} \right)}} \\{{\psi \left( {{- v}\; {\tau_{r}/N}} \right)}{\phi \left( {{\tau + {\tau_{r}/N}},v} \right)}} \\\vdots \\{{\psi \left( {{- v}\; {\tau_{r}\left( {1 - {1/N}} \right)}} \right)}{\phi \left( {{\tau + {\tau_{r}\left( {1 - {1/N}} \right)}},v} \right)}}\end{bmatrix}},$

for every φ∈

(V,π_(ϵ)) and (τ,v)∈V.

B4. ZDMA Transceiver Embodiments

In this section we describe the structure of the ZDMA transceiverincorporating the Zak realization formalism. In addition, we describe aweaker version that can be implemented as a preprocessing step overmulti-carrier modulation.

B4.1 Transceiver Parameters

The ZDMA transceiver structure is based on the following parameters:

-   -   (I) An Heisenberg critically sampled lattice (ƒ,ϵ) giving rise        to the Hilbert space of Zak waveforms        (V,π_(ϵ)).    -   (2) A transmit and receive filter functions w_(tx),w_(rx)∈        (V).    -   (3) A non-degenerate pulse waveform φ∈        (V,π_(ϵ)) satisfying P(v)≠0 for every v∈V.

The transmit function w_(tx) is a function on the delay Doppler planethat plays the role of a two dimensional filter, shaping the transmittedsignal to a specific bandwidth and specific time duration. The receivefunction w_(rx) is principally the matched filter to w_(tx). We willassume, henceforth that it is defined as w_(rz)=w_(tx) ^(★):

$\begin{matrix}\begin{matrix}{{w_{sx}(v)} = {w_{sx}^{\bigstar}(v)}} \\{{= {{\psi \left( {- {\beta \left( {v,v} \right)}} \right)}\overset{\_}{w_{tx}\left( {- v} \right)}}},}\end{matrix} & (4.1)\end{matrix}$

for every v∈V. In addition, we assume that the function w=w_(tx) can bedecomposed as a twisted/Heisenberg convolution w=w_(γ)

w_(v) where w_(τ) is a one dimensional function supported on the delayaxis and w_(v) is one dimensional function supported on the Doppleraxis. One can verify that:

(w(τ,v)=w _(γ)(γ)w _(v)(v)  (4.2),

for every (τ,v)∈

²=V. The benefit in considering such decomposable filters is that mostof their attributes can be expressed in terms of simple analysis of thetwo factors. In particular, in this situation the received matchedfunction is given by w_(rx) ^(★)=w_(v) ^(★)

w₄ ^(★) where w_(r) ^(★) and w_(v) ^(★) are the respected onedimensional conjugate functions familiar from standard signalprocessing, i.e.,:

w _(r) ^(★)(τ)= w _(r)−γ),  (4.3)

w _(v) ^(★)(v)= w _(v)(−v),  (4.4)

for every τ∈

and v∈

respectively. In typical situation, we require the one dimensionalfilter w_(τ) to be a square root Nyquist with respect to a bandwidthB>0, i.e., w_(τ) ^(★)*w_(τ)(k/B)=0 for every non-zero integer k, and,reciprocally, we require the one dimensional filter w_(v) to be squareroot Nyquist with respect to a duration T>0, i.e., w_(v)^(★)*w_(v)(l/T)≠0 for every non-zero integer l. To proceed further weneed to choose a basis of the lattice Λ:

Λ=

g ₁⊕

g ₂,  (4.5)

Granting such a choice we can define the pulse P to be the uniquequasi-periodic function that satisfy P(ag₁+bg₂) for every 0≤a,b≤1. Notethat when the lattice is rectangular and the basis is the standard one,such pulse is described in Example 2.2. Before describing thetransceiver structure we need to explain how to encode the informationbits. These are encoded into a periodic function x∈

(V) with respect to the lattice Λ. In typical situations we assume thatx is a sampled Λ—periodic function of the form:

$\begin{matrix}{{x = {\sum\limits_{n,m}{{x\left\lbrack {n,m} \right\rbrack}{\delta \left( {{{ng}_{1}\text{/}N} + {{mg}_{2}\text{/}M}} \right)}}}},} & (4.6)\end{matrix}$

where N,M∈

≥¹ are fixed parameters defining the density of the delay Dopplerinformation lattice Λ_(N,M)=

g₁N⊕

g₂/M. In more canonical terms, x is a function on the informationlattice. Λ_(N,M) that is periodic with respect to the sub-latticeΛ⊂ƒ_(N,M). The expression for x is particularly simple when the latticeis rectangular. In this case it takes the form:

$\begin{matrix}{{x = {\sum\limits_{n,m}{{x\left\lbrack {n,m} \right\rbrack}{\delta \left( {n\; \tau_{\tau}\text{/}N} \right)}{\delta \left( {{mv}_{\tau}\text{/}M} \right)}}}},} & (4.7)\end{matrix}$

B4.2 Transceiver structureHaving specified the underlying structures andparameters we are now ready to describe the ZDMA modulation andde-modulation transforms. Given a periodic function x∈

(V/Λ) encoding the information bits, we define the modulated waveformφ_(tx)=

(x) through the rule:

$\begin{matrix}\begin{matrix}{{\mathcal{M}(x)} = {_{{time},\epsilon}\left( {{\Pi^{\epsilon}\left( w_{tx} \right)} \vartriangleright \left( {x \cdot P} \right)} \right)}} \\{= {_{{time},\epsilon}\left( {w_{tx}\left( {x \cdot P} \right)} \right)}}\end{matrix} & (4.8)\end{matrix}$

where

_(time,ϵ); is the Zak transform converting between Zak and time domainwaveforms, see (2.7). In words, the modulation first transforms theinformation function into a Zak waveform through multiplication bymultiplication with P. Next, it shapes the bandwidth and duration of thewaveform through twisted convolution with the two dimensional filter w.Last, it transforms the tamed Zak waveform into a time domain waveformthrough application of the Zak transform. To get a better understandingof the structure of the transmit waveform and the actual effect of twodimensional filtering we apply several algebraic manipulations to getmore concrete expressions. First, we note that

_(time,ϵ); is an intertwining transform thus obeying the relation:

_(time,ϵ)(Π^(ϵ)(w _(tx))

(x·P))=Π^(time)(w _(tx))

_(time,ϵ)(x·P),  (4.9)

Second, assuming w_(tx)=w_(τ)

w_(v), we can write Π^(time)(w_(r)

w_(v)) as the composition time Π^(time)(w_(r))○Π^(time) (w_(v)), thusexpressing the two dimensional filtering operation as cascade of twoconsecutive one dimensional operations:

$\begin{matrix}\begin{matrix}{{\mathcal{M}(x)} = {{\Pi^{time}\left( {x_{\tau}w_{v}} \right)} \vartriangleright {_{{time},\epsilon}\left( {x \cdot P} \right)}}} \\{= {{\Pi_{time}\left( w_{\tau} \right)} \vartriangleright {\Pi_{time}\left( w_{v} \right)} \vartriangleright {_{{time},\epsilon}\left( {x \cdot P} \right)}}} \\{{= {w_{\tau}*\left\{ {W_{t} \cdot {_{{time},\epsilon}\left( {x \cdot P} \right)}} \right\}}},}\end{matrix} & (4.10)\end{matrix}$

where * stands for linear convolution on

and W_(l)=FT⁻¹(w_(v)). We refer to the waveform

_(time,ϵ)(x·P) as the bare ZDMA waveform. We see from Formula (4.10)that the transmit waveform is obtained from the bare waveform byapplying the time window W_(t). followed by a convolution with the pulsew_(τ). In addition, one can verify that when x is sampled on the latticeΛ_(N,M)⊂V, see (4.6), the bare waveform is an infinite delta pulse trainalong the lattice:

ζ_(N,M) ^(time) ={n/Ng ₁[1]+m/Mg ₂[1]:n,m∈

},  (4.11)

where the lattice Λ_(N,M) ^(time) is the projection of the latticeΛ_(N,M) on the delay axis. The projected lattice takes a particularlysimple form when Λ=Λ_(τ) is rectangular. In this case we have:

Λ_(N,M) ^(time)=

τ_(γ) /N  (4.12)

We now proceed to describe the de-modulation mapping. Given a receivedwaveform φ_(rx) we define its de-modulated image y=

(φ_(rx)) through the rule:

(φ_(rx))=Π^(ϵ))w _(rx))

_(ϵ,time)(φ_(rx)),  (4.13)

We stress the fact that y is a Zak waveform (to be distinguished from aperiodic function). We often incorporate as part of the de-modulationmapping another step of sampling y on the lattice Λ_(N,M), thusobtaining a sampled Zak waveform y_(ϵ)∈

(Λ_(N,M),π_(ϵ)) which is a function on Λ_(N,M) satisfying the quasiperiodicity condition:

$\begin{matrix}{{y_{\epsilon}\left( {\delta + \lambda} \right)} = {{\psi \left( {\beta \left( {\delta,\lambda} \right)} \right)}{\pi_{\epsilon}(\lambda)}^{- 1}\mspace{11mu} \mspace{11mu} {{y_{\epsilon}(\delta)}.}}} & (4.14)\end{matrix}$

To conclude, assuming x=x∈

(Λ_(N,M)/79 ) is a sampled periodic function, the ZDMA transceiver chainconverts it into a sampled Zak waveform y

∈

(ƒ_(N,M),π_(ϵ)). Overall, the composition transform

○

is a linear transformation:

○

:

(⊕_(N,M)/Λ)→

(Λ_(N,M),π_(ϵ)),  (4.15)

taking sampled periodic functions to sampled Zak waveforms. Inparticular, we have that both the domain and range of

○

are finite dimensional vector spaces of dimension N·M. In the nextsubsection we will analyze in more detail the exact relation between theinput variable x and the output variable y.

B4.3 Input output relation We first assume the channel between thetransmitter and receiver is trivial. Furthermore, we assume the receivedfilter is matched to the transmit filter, i.e., w_(rx)=w_(tx) ^(★). Forsimplicity we denote w=w_(tx) and assume w is decomposable, i.e.,w=w_(τ)

w_(v). At this stage we do not assume anything about the specificstructure of the one dimensional filters w_(γ) and w_(v). Given an inputfunction x∈

(V/Λ), a direct computation reveals that y=D○

(x) is given by:

$\begin{matrix}\begin{matrix}{y = {{\Pi^{\epsilon}\left( w^{\bigstar} \right)} \vartriangleright {\Pi^{\epsilon}(w)} \vartriangleright \left( {x \cdot P} \right)}} \\{= {{\Pi^{\epsilon}\left( {w^{\bigstar}w} \right)} \vartriangleright \left( {x \cdot P} \right)}} \\{{= {\left( {w^{\bigstar}w} \right)\left( {x \cdot P} \right)}},}\end{matrix} & (4.16)\end{matrix}$

So we see thaty is given by the twisted convolution of x×P by theauto-correlation filter w^(★)

w. Our goal is to calculate an explicit formula for w^(★)

w. First we note that since w^(★)=w_(v) ^(★)

w_(τ) ^(★), we can write:

$\begin{matrix}\begin{matrix}{{w^{\bigstar} \otimes w} = {w_{v}^{\bigstar}w_{\tau}^{\bigstar}w_{\tau}w_{v}}} \\{{= {w_{v}^{\bigstar}w_{\tau}^{(2)}w_{v}}},}\end{matrix} & (4.17)\end{matrix}$

where w_(τ) ⁽²⁾=w_(τ) ^(★)

w_(τ) is the one dimensional auto-correlation function of the delayfilter w_(t). In addition, since w_(τ) ⁽²⁾ is supported on the τ axisand w_(v) ^(★) is supported on v axis, we have the following simplerelation:

$\begin{matrix}\begin{matrix}{{w_{v}^{\bigstar}{w_{\tau}^{(2)}\left( {\tau,v} \right)}} = {{w_{\tau}^{(2)}(r)}\left( {\tau \; v} \right){w_{v}^{\bigstar}(v)}}} \\{{= {{w_{\tau}^{(2)}(r)}{M_{\tau}\left\lbrack w_{v}^{\bigstar} \right\rbrack}(v)}},}\end{matrix} & (4.18)\end{matrix}$

Thus, for any given point (τ,v) we can write w^(★)

w(τ,v) in the form:

w ^(★)

w(τ,v)=w _(τ) ⁽²⁾(τ)⊗{tilde over (w)} _(v) ⁽²⁾(v)  (4.19)

where {tilde over (w)}_(v) ⁽²⁾=M_(τ)[w_(v) ^(★)]

w_(v). We note that the definition of {tilde over (w)}_(v) ⁽²⁾ dependson the point τ which is not apparent from the notation we use. Formula(4.19) is always true and is pivotal to get approximations of w^(★)

w under various assumptions on the filters w_(τ) and w_(v). The case ofinterest for us is when w_(τ) and w_(v) are square root Nyquist withrespect to a bandwidth B and a duration T∈

≥0 respectively and, in addition, B·T≥≥1. In this case we canapproximate {tilde over (w)}_(v) ^(★)(v)=ψ(τv)w_(v) ^(★)(v)˜w_(v)^(★)(v), thus {tilde over (w)}_(v) ⁽²⁾˜w_(v) ⁽²⁾. which in turns implythat:

w ^(★)

w˜107 _(τ) ⁽²⁾

w _(v) ⁽²⁾,  (4.20)

In particular, in the rectangular situation where Λ=Λ_(r),B=Nv_(r) andT=Mτ_(r) such that NM>>1, placing the QAM symbols on the latticeƒ_(N,M)=

(τ_(r)/N,0)⊕

(0, v_(r)/M) yields:

$\begin{matrix}\begin{matrix}{y = {{\Pi^{\epsilon}\left( {w^{\bigstar}w} \right)} \vartriangleright \left( {x \cdot P} \right)}} \\{{{\sim {x \cdot P}},}}\end{matrix} & (4.21)\end{matrix}$

thus allowing perfect reconstruction without equalization when thechannel is AWGN. Next, we describe the input-output relation in thepresence of a non-trivial channel H=Π^(time)(h) where h=h(τ,v) is thedelay Doppler impulse response. For the sake of the analysis, it issufficient to assume that h is a single reflector, i.e.,

h(τ,v)=δ(τ−τ₀, v−v₀)=δ(τ−τ₀)

δ(v−v₀). In the following computation we use the short notationsδ_(τ0)=δ(τ−τ₀) and δ_(v0)=δ)v−v₀). Given an input x∈

(V/Ε), a direct computation reveals that the transmit-receive image y=

○H○

(x) is given by:

y=Å ^(ϵ)(w ^(★)

h

)

(x·P),  (4.22)

Our goal is to calculate an explicit formula for w^(★)

w. To do that, we first write:

$\begin{matrix}\begin{matrix}{{w^{\bigstar}hw} = {w_{v}^{\bigstar}w_{\tau}^{\bigstar}\delta_{\tau_{0}}\delta_{\tau_{0}}w_{\tau}w_{v}}} \\{= {w_{v}^{\bigstar}{L_{\tau_{0}}\left\lbrack w_{\tau}^{\bigstar} \right\rbrack}{M_{v_{0}}\left\lbrack w_{\tau} \right\rbrack}{L_{v_{0}}\left\lbrack w_{v} \right\rbrack}}} \\{{= {w_{v}^{\bigstar}{L_{\tau_{0}}\left\lbrack {\overset{\sim}{w}}_{\tau}^{(2)} \right\rbrack}{L_{v_{0}}\left\lbrack w_{v} \right\rbrack}}},}\end{matrix} & (4.23)\end{matrix}$

where {tilde over (w)}_(γ) ⁽²⁾=w_(τ) ^(★)

M_(v0)[w_(τ)]. In addition, we have:

$\begin{matrix}\begin{matrix}{{w_{v}^{\bigstar}{L_{\tau_{0}}\left\lbrack {\overset{\sim}{w}}_{\tau}^{(2)} \right\rbrack}\left( {\tau,v} \right)} = {{L_{\tau_{0}}\left\lbrack {\overset{\sim}{w}}_{\tau}^{(2)} \right\rbrack}{(\tau) \otimes {\psi \left( {\tau \; v} \right)}}{w_{v}^{\bigstar}(v)}}} \\{{= {{L_{\tau_{0}}\left\lbrack {\overset{\sim}{w}}_{\tau}^{(2)} \right\rbrack}{(\tau) \otimes {M_{\tau}\left\lbrack w_{v}^{\bigstar} \right\rbrack}}(v)}},}\end{matrix} & (4.24)\end{matrix}$

Hence, overall we can write:

w ^(★)

h

w=h*w _(τ) ⁽²⁾ *w _(v) ⁽²⁾  (4.25)

where {tilde over (w)}_(v) ⁽²⁾=M_(v0)[w_(v) ^(★)]

w_(v). If we assume that and are Nyquist with respect to a bandwidth Band a duration T respectively, and, in addition have B·T>>1 and v₀>>Bthen we can approximate {tilde over (w)}_(τ) ⁽²⁾˜w_(τ)^((2) and {tilde over (w)}) _(v) ⁽²⁾, which, in turns, imply:

w ^(★)

h

w=h*w _(τ) ⁽²⁾ *w _(v) ⁽²⁾,  (4.26)

B4.4 Channel acquisitionLooking back at the input output relationy=h_(w)

(x·P) where h_(w=w) ^(★)

w, we proceed to derive a simple acquisition scheme of the filteredchannel impulse response h_(w). To this end, we fix a point v₀∈V andconsider the standard pulse structure P(ag₁−bg₂)=1 for 0≤a,b≤1. Giventhese choices, we define the pilot structure as the Zak waveform φ

B4.5 Weak ZDMA. In this subsection, we describe a weak variant of theZDMA transceiver that can be architected as a pre-processing layer overmulti-carrier transceiver. We refer to this transceiver as w-ZDMA. Thedefinition of the w-ZDMA transceiver depends on similar parameters asthe ZDMA transceiver we described before however, with few additionalassumptions. First assumption is that the transmit and receive filtersare periodic with period Λ, i.e., w_(tx), w_(rx)∈

(V/Λ). in other words, w_(tx/rx)=SF(W_(tx/rx)) where w_(tx), w_(rx)∈

(V/Λ) are discrete window functions and SF is the symplectic Fouriertransform. The support of the window W_(tx) determines the bandwidth andduration of the transmission packet. Typically, we take the receivefilter to be matched W_(rx)=W _(tx) or, equivalently, that w_(rx)=w_(tx)^(★) Another assumption is that the generator signal P∈

(V,π_(ϵ)) satisfy the orthogonality condition:

P·P=1  (4.27)

We note that condition (4.27) is equivalent to Gabor orthogonalitycondition of the waveform p=

_(time,ϵ)(P) with respect to the Heisenberg lattice Im{circumflex over(ϵ)}={(λ,ϵ(λ)):λ∈Λ}. To see this, let λ∈Λ and consider the inner product\

p,π^(time)(λ,ϵ(λ))

p

.

Now write:

$\begin{matrix}\begin{matrix}{{\langle{p,{{\pi^{time}\left( {\lambda,{\epsilon (\lambda)}} \right)} \vartriangleright p}}\rangle} = {\langle{{_{{time},e}(P)},{{\pi^{time}\left( {{\lambda\epsilon}(\lambda)} \right)} \vartriangleright {_{{time},e}(P)}}}\rangle}} \\{= {\langle{{_{{time},e}(P)},{_{{time},e}\left( {{\pi^{\epsilon}\left( {\lambda,{\epsilon (\lambda)}} \right)} \vartriangleright P} \right)}}\rangle}_{t}} \\{= {\int_{V\text{/}A}{{\overset{\_}{P}(v)}\left\{ {{\pi^{e}\left( {\lambda,{\epsilon (\lambda)}} \right)} \vartriangleright P} \right\} (v){dv}}}} \\{= {\int_{V\text{/}A}{{\psi \left( {w\left( {\lambda,v} \right)} \right)}{\overset{\_}{P}(v)}{P(v)}{dv}}}} \\{= {\int_{V\text{/}A}{{{\psi \left( {w\left( {\lambda,v} \right)} \right)} \cdot 1}{dv}}}} \\{= \left\{ {\begin{matrix}1 & {\lambda = 0} \\0 & {\lambda \neq 0}\end{matrix},} \right.}\end{matrix} & (4.28)\end{matrix}$

where in the second equality we use the fact

_(time,ϵ) is an intertwining transform and in the fourth equality we use(2.5). Given an input function x∈

(V/Λ) encoding the information bits, we shape the band and duration ofthe transmitted waveform through periodic convolution with w_(tx). i.e.,x

w_(tx)*x. Overall, the signal x is modulated according to the followingrule:

(x)=

_(time,ϵ)((w _(tx)*x)·P),  (4.29)

It is illuminating to compare Formula (4.29) with Formula (4.8). Oneobserves that the main difference is in the way the shaping filter isapplied where in ZDMA it is applied through the operation of twistedconvolution x

w_(tx)

(x·P) and in w-ZDMA through the operation of periodic convolution x

(w_(tx)*x)·P. We next explain how the modulation rule (4.29) can beexpressed as a layer over multi-carrier modulation scheme. To this end,let us write w*x in the form w*x=SF(W·X) where x=SF(X), that is:

$\begin{matrix}{{{w*{x(v)}} = {\sum\limits_{\lambda \in \Lambda}{{\psi \left( {- {w\left( {v,\lambda} \right)}} \right)}{{W(\lambda)} \cdot {X(\lambda)}}}}},} & (4.30)\end{matrix}$

For every v∈V. Hence:

$\begin{matrix}\begin{matrix}{{_{{time},e}\left( {\left( {w*x} \right) \cdot P} \right)} = {\sum\limits_{\lambda \in \Lambda}{{{W(\lambda)} \cdot {X(\lambda)}}{_{{time},e}\left( {{\psi \left( {- {w\left( {v,\lambda} \right)}} \right)}P} \right)}}}} \\{= {\sum\limits_{\lambda \in \Lambda}{{{W(\lambda)} \cdot {X(\lambda)}}{_{{time},e}\left( {{\psi \left( {w\left( {v,\lambda} \right)} \right)}P} \right)}}}} \\{= {\sum\limits_{\lambda \in \Lambda}{{{W(\lambda)} \cdot {X(\lambda)}}{_{{time},e}\left( {{\pi^{e}\left( {\lambda,{\epsilon (\lambda)}} \right)} \vartriangleright P} \right)}}}} \\{= {{\sum\limits_{\lambda \in \Lambda}{{{W(\lambda)} \cdot {X(\lambda)}}{\pi^{time}\left( {\lambda_{s}{\epsilon (\lambda)}} \right)}}} \vartriangleright {_{{time},e}(P)}}} \\{= {{\sum\limits_{\lambda \in \Lambda}{{{W(\lambda)} \cdot {X(\lambda)}}\pi^{time}\left( {\lambda_{s}{\epsilon (\lambda)}} \right)}} \vartriangleright p}} \\{{= {{\Pi^{time}\left( {{\hat{\epsilon}}_{pf}\left( {W \cdot {{SF}(x)}} \right)} \right)} \vartriangleright p}},}\end{matrix} & (4.31)\end{matrix}$

In case the Heisenberg character ϵ=1. e.g., when Λ=Λ_(γ) is rectangular,the modulation formula becomes

(x)=Π^(time)(W·SF(x))

p. We now proceed to describe the de-modulation rule implemented at thereceiver side. Given a time domain waveform φ_(rx), the receiverdemodulates it according to the following formula:

(φ_(rx))=

_(ϵ,time)(φ_(rx))· P,  (4.32)

Observe that when the channel is identity, due to the orthogonalitycondition (4.27), we obtain perfect reconstruction after composingmodulation and demodulation:

$\begin{matrix}\begin{matrix}{{\; o\; {\mathcal{M}(x)}} = {{x \cdot P}\overset{\_}{P}}} \\{= {x \cdot 1}} \\{{= x},}\end{matrix} & (4.33)\end{matrix}$

We further note that the orthogonality condition is not essential forachieving perfect reconstruction. In fact, one needs to impose is that Pis non-degenerate, that is, that PP is nowhere vanishing. Fornon-degenerate P one can reconstruct the input x as:

x=

○

(x)/PP,

The use of general non-degenerate generator functions give rise tonon-orthogonal variants of the w-ZDMA transceiver. For a non-trivialchannel transformation of the form H=π^(time)(v₀) where v₀=(τ₀,v₀) weget:

$\begin{matrix}{\begin{matrix}{y = {{{o}\; {Ho}\; {\mathcal{M}(x)}} = {\left( {{\pi^{time}\left( \upsilon_{0} \right)} \vartriangleright {\mathcal{M}(x)}} \right)}}} \\{= {{_{\epsilon,{time}}\left( {{\pi^{time}\left( \upsilon_{0} \right)} \vartriangleright {\mathcal{M}(x)}} \right)} \cdot \overset{\_}{P}}} \\{= {\left\{ {{\pi^{\epsilon}\left( \upsilon_{0} \right)} \vartriangleright {_{\epsilon,{time}}\left( {\mathcal{M}(x)} \right)}} \right\} \cdot \overset{\_}{P}}} \\{{= {\left\{ {{\pi^{\epsilon}\left( \upsilon_{0} \right)} \vartriangleright {x \cdot P}} \right\} \overset{\_}{P}}},}\end{matrix}\quad} & (4.34)\end{matrix}$

where the second equality is the definition of

, the third equality is the intertwining property of the Zak transformand the last equality is the definition of

. Finally, let v=(τ,v) where 0≤τ,v<1 and evaluate y at v:

y(v)=ψ(−β(v ₀ ,v ₀)ψ(β(v ₀ ,v))x(v−v ₀)P(v−v ₀ P (v)  (4.35)

Assuming v₀ is small compared to the lattice dimensions and that P is acontinuous function we get the approximation (The continuity assumptionof G does not always hold, for example in the most common case when G=

⁻¹(1_([G,T])) the case of the standard window function.)

$\begin{matrix}{\begin{matrix}{{y(\upsilon)} \simeq {{x\left( {\upsilon - \upsilon_{0}} \right)}{P(\upsilon)}{\overset{\_}{P}(\upsilon)}}} \\{{= {x\left( {\upsilon - \upsilon_{0}} \right)}},}\end{matrix}\quad} & (4.36)\end{matrix}$

where for the approximation we used the fact that ψ(β(v₀v)),ψ(−β(v₀,v₀))≃1 and that P(v−v₀)≃P(v) by continuity. Note that when P correspondsto the standard window (see Example 2.2) the approximation (4.36) is nolonger valid since P is not continuous at the boundaries of thefundamental cell.

C0. Introduction to Radar Waveform Design in the Zak Realization

In the subsequent sections, a general systematic method for radarwaveform design that is based on the Zak representation of discretesequences and continuous signals (aka waveforms) is described. Along theway we develop the theory of sampling and filtering using the formalismof the Heisenberg group. We conclude with an example of a particularfamily of compressed radar waveforms based on discrete Zak sequences.These waveforms enjoy uniform temporal power profile and thumb-tack likeambiguity function with a clean punctured region around the origin whosedimensions are free parameters.

C1. Set-up for Radar Waveform Design

Let V=

² be the delay Doppler plane equipped with the standard symplectic formw:

ω(v ₁ ,v ₂)=v ₁τ₂ −v ₂τ₁,

for every v₁=(τ₁,v₁) and v₂=(τ₂,v₂). Let β be the polarization form:

β(v ₁ ,v ₂)=v ₁τ₂.

Using the form β we introduce a binary operation between functions on Vcalled twisted convolution. To simplify notations, we denote byψ(z)=exp(j2πz) the standard Fourier exponent. Given a pair of functionsh₁,h₂∈

(V), we define their twisted convolution to be:

h₁*_(σ)h₂(v) = ∫_(v₁ + v₂ = v)(β(v₁, v₂))h₁(v₁)h₂(v₂),

We fix a critically sampled lattice Λ₁⊂V. We assume Λ₁ ⊃V. We assume Λ₁is rectangular of the form:

Λ₁=

τ_(r) ⊕

Δv _(r),

such that τ_(r)·v_(r)=1. We fix a rectangular super-lattice Λ⊂Λ₁ of theform:

Λ=

Δτ⊕

Δv,

where Δτ=τ_(r)/N and Δv=v_(r)/M. We denote by L=[Λ:Λ₁] the index of Λ₁as a sub-lattice of Λ. It is easy to verify that L=N·M. This number alsocounts the number of points in the finite quotient group Λ/Λ₁ . Inaddition, we denote by Λ^(⊥) the symplectic orthogonal complement of Λdefined by:

Λ^(⊥) ={v∈V:w(v,λ)∈

for every λ∈Λ},

We have Λ^(⊥⊂Λ) ₁. Overall, we defined a nested family of lattices (seeFIG. 10):

Λ^(⊥⊂Λ) ₁⊂Λ,

We note that [Λ:Λ^(⊥)]=L² or, equivalently, the number of points in thequotient group Λ/Λ^(⊥) is equal to L² . Finally, we introduce a discretevariant of the twisted convolution operation between functions on thelattice Λ. Given a pair of functions h₁,h₂∈

(ƒ), we define their twisted convolution to be:

h₁*_(σ)h₂(v) = ∫_(λ₁ + λ₂ = λ)ψ(β(λ₁, λ₂))h₁(λ₁)h₂(λ₂).

C2. Continuous Zak Signals

In classical signal processing there are two fundamental domains ofsignal realizations: the time domain and the frequency domain. Each ofthese domains reveal complementary attributes and the conversion betweenthese two realizations is carried through the Fourier transform. As itturns out, there is another fundamental domain called the Zak domain. Acontinuous Zak signal is a function Φ:V→

that satisfies the quasi-periodicity condition:

Φ(v+λ ₁)=ψ(62 (v,λ ₁))Φ(v),

for every v∈V and λ₁∈Λ₁. Concretely, if we take v=(τ,v) andλ₁=(kτ_(r),lv_(r)) then condition (2.1) takes the form:

Φ(τ+kτ _(r) ,v+lv _(r))=ψ(kvτ _(r))Φ(τ,v),

Given a pair of Zak signals Φ₁Φ₂∈

we define their inner product as:

${{\langle{\Phi_{1},\Phi_{2}}\rangle} = {\int_{V/\Lambda_{1}}~{{\overset{\_}{\Phi_{1}(v)} \cdot {\Phi_{2}(v)}}{dv}}}},$

We denote the Hilbert space of continuous Zak signals by

=

(V/Λ₁,β). We equip

with an Heisenberg action defined by the operator valued transform Π:

(V)→End(

) defined by:

Π(h)

Φ=h* _(σ)Φ,

for every Φ∈

and h∈

(V). We refer to Π as the Heisenberg transform. The Heisenberg transformadmits an inverse called the Wigner transform. Given a pair of Zaksignals Φ₁,Φ₂∈

, the Wigner transform of the rank one operator |Φ₂

Φ₁| is the function

_(Φ) ₁ _(,Φ) ₂ :V→

given by:

₁₀₁ ₁ _(Φ) ₂ (v)=

π(v)Φ₁,Φ₂

,

for every v∈V, where π(v)=Π(δ(v)). The function (2.5) is called thecross-ambiguity function of the signals Φ₁ and Φ₂. In case Φ₁=Φ₂=Φ, wedenote the cross-ambiguity function simply by

_(Φ) and refer to it as the ambiguity function of the signal Φ. Theconversion between the Zak domain to the time domain is carried throughthe Zak transform

:

→L²(t∈

), given by:

(Φ) = ∫₀^(v_(r))Φ (τ, v)dv,

for every Φ∈

. We conclude this section with an example of an explicit Zak signal andits time domain realization. Let Φ_(n,m) be the unique quasi-periodicextension of the delta function supported on the lattice point(nΔτ,mΔv), for 0≤n≤N−1 and 0≤m≤M−1, i.e.,:

$\begin{matrix}{\Phi_{n,m} = {\sum\limits_{k,l}{{\psi \left( {m\; \Delta \; {v \cdot k}\; \tau_{r}} \right)}{\delta \left( {{{n\; \Delta \; \tau} + {k\; \tau_{r}}},{{m\; \Delta \; v} + {lv}_{r}}} \right)}}}} \\{= {\sum\limits_{k,l}{{\psi \left( {{mkv}_{r} \cdot {\tau_{r}/M}} \right)}{\delta \left( {{{n\; \Delta \; \tau} + {k\; \tau_{r}}},{{m\; \Delta \; v} + {lv}_{r}}} \right)}}}} \\{{= {\sum\limits_{k,l}{{\psi \left( {{mk}/M} \right)}{\delta \left( {{{n\; \Delta \; \tau} + {k\; \tau_{r}}},{{m\; \Delta \; v} + {lv}_{r}}} \right)}}}},}\end{matrix}\quad$

Direct calculation reveals that the Zak transform of Φ_(n,m) is a timeshifted, phase modulated, infinite delta pulse train (see FIG. 5), givenby:

${{\left( \Phi_{n,m} \right)} = {\sum\limits_{k \in {\mathbb{Z}}}{{\psi \left( {{mk}/M} \right)}{\delta \left( {{n\; \Delta \; \tau} + {k\; \tau_{r}}} \right)}}}},$

C3. Discrete Zak Signals

The continuous Zak theory admits a (finite) discrete counterpart whichwe proceed to describe. The development follows the same lines as in theprevious section. We use lower case letters to denote discrete Zaksignals. A discrete Zak signal is a function ϕ:Λ→

that satisfies the following quasi-periodicity condition:

ϕ(λ++₁)=ψ(β(λ,λ₁))ϕ(λ),

for every λ∈ζand λ₁∈Λ₁. Concretely, if we take λ=(nΔτ,mΔv) andλ₁=(kτ_(γ), lv_(r)) then condition (3.1) takes the form:

$\begin{matrix}{{\varphi \left( {{{n\; \Delta \; \tau} + {k\; \tau_{r}}},{{m\; \Delta \; v} + {lv}_{r}}} \right)} = {{\psi \left( {{mk}\; \Delta \; v\; \tau_{r}} \right)}{\varphi \left( {{n\; \Delta \; \tau},{m\; \Delta \; v}} \right)}}} \\{= {{\psi \left( {{mkv}_{r}{\tau_{r}/M}} \right)}{\varphi \left( {{n\; \Delta \; \tau},{m\; \Delta \; v}} \right)}}} \\{{= {{\psi \left( {{mk}/M} \right)}{\varphi \left( {{n\; \Delta \; \tau},{m\; \Delta \; v}} \right)}}},}\end{matrix}\quad$

Given a pair of discrete Zak signals ϕ₁,ϕ₂∈

_(L) we define their inner product as:

${{\langle{\Phi_{1},\Phi_{2}}\rangle}_{L} = {\sum\limits_{\lambda \in {\Lambda/\Lambda_{1}}}~{\overset{\_}{\varphi_{1}(\lambda)} \cdot {\varphi_{2}(\lambda)}}}},$

We denote the Hilbert space of discrete Zak signals by

_(L)=

(Λ/Λ₁,β). One can show that dim

_(L)=L. We equip

_(L) with an action of the finite Heisenberg group, expressed throughthe transform Π_(L):F(Λ)→End(

_(L)), given by:

Π_(L)(h)

ϕ=h* _(σ)ϕ,

for every ϕ∈

_(L) and h∈

(Λ) . We refer to Π_(L) as the discrete Heisenberg transform. Thediscrete Heisenberg transform admits an inverse called the discreteWigner transform. Given a pair of discrete Zak signals Φ₁Φ₂ ∈

_(L), the discrete Wigner transform of the rank one operator |ϕ₂

ϕ₁| is the function

_(ϕ) ₁ _(ϕ) ₂ :Λ→

given by:

_(ϕ) ₁ _(ϕ) ₂ (λ)=

π(λ)

ϕ₁,ϕ₂

_(L) ,

for every λ∈Λ, where π_(L)(λ)=Π_(L)(δ(λ)). The function (3.5) is calledthe discrete cross-ambiguity function of the signals ϕ₁ and ϕ₂. Sinceπ_(L)(λ−λ^(⊥))=π_(L)(λ) for every λ∈Λandλ^(⊥), it follows that

_(,ϕ) ₁ _(ϕ) ₂ is a periodic with respect to the sub-lattice Λ^(⊥),i.e.,:

_(ϕ) ₁ _(,ϕ) ₂ (λ+λ^(⊥))=

_(ϕ) ₁ _(,ϕ) ₂ (λ),

for every λ∈Λ and λ^(⊥)∈Λ^(⊥). When ϕ₁=ϕ₂=ϕ we denote the discretecross-ambiguity function by

_(ϕ) and refer to it as the discrete ambiguity function of ϕ.

C4. Sampling Theory on the Zak Domain

The focus of sampling theory is to describe the relation between thecontinuous and discrete cross-ambiguity functions. To this end, wedenote by

′(V) the vector space of generalized functions on V. The main assertionsare stated in terms of two basic transforms:

s:

(V)-→

(Λ),

l:

(Λ)-→

(V),

The transform s is called sampling and it sends a function on V to itssamples on the lattice Λ. The transform l is called embedding and itsends a discrete function ϕ:Λ→

to the generalized function (distribution) on V given by the followingsuper-position of delta functions:

${{\iota (\varphi)} = {\sum\limits_{\lambda \in \Lambda}~{{\varphi (\lambda)}{\delta (\lambda)}}}},$

The sampling and embedding transforms give rise to induced transformsbetween the corresponding Hilbert spaces of continuous and discrete Zaksignals. We denote the induced transforms by the same names, i.e.,:

s:

-→

_(L),

l:

_(L)-→

′,

where

′=

′(V/Λ₁β) denotes the vector space of generalized Zak signals(distributions). Given a function h∈

(Λ), we denote by h_(Λ) ^(⊥) its periodization with respect to thesub-lattice Λ^(⊥⊂)Λ, i.e.,:

${{h_{\Lambda^{\bot}}(\lambda)} = {\sum\limits_{\lambda^{\bot} \in \Lambda^{\bot}}{h\left( {\lambda + \lambda^{\bot}} \right)}}},$

for every λ∈Λ. The main technical statement is summarized in thefollowing theorem.

Theorem 4.1 (Main Theorem of Sampling Theory). The following tworelations hold:

(1) Sampling relation. For every Φ₁,Φ₂ ∈

we have:

_(s(Φ) ₁ _(),s(Φ) ₂ ₎ =s(

_(Φ) ₁ _(,Φ) ₂ )_(Λ) _(⊥) ,

(2) Embedding relation. For every Φ₁, Φ₂∈

_(L) we have:

_(l(ϕ) ₁ _(),l(ϕ) ₂ ₎ =l(

_(ϕ) ₁ _(,ϕ2)),

In plain language, the sampling relation asserts that the discretecross-ambiguity function of sampled continuous signals is the sampled(and periodized) cross-ambiguity function of the continuous signals. Theembedding relation asserts that the continuous cross-ambiguity functionof embedded discrete signals is the embedding of the cross-ambiguityfunction of the discrete signals.

C5. Filter Theory

Filter theory gives means to convert a discrete sequences to continuouswaveforms. We define an Heisenberg filter to be a function w∈

(V). We say the filter w is factorizable if it can be written asw=w_(τ)*_(σ)w_(v) where w_(τ) is a distribution supported on the delayaxis and w_(v) is a distribution supported on the Doppler axis. Notethat such a function takes the form:

w(τ,v)=w _(τ)(τ)w _(v)(v),

for every τ,v∈

. The manner of operation of a filter w on a Zak signal Φ is carriedthrough the Heisenberg transform, i.e.,:

Φ_(w)=Π(w)

Φ=w* _(σ)Φ,

The above equation shows a relationship between Zak signal and theHeisenberg transform. While the relationship is described as a sequenceof mathematical steps, in general, implementations need not explicitlyperform these steps, but may use numerical methods to compute endresults without having to compute and store any intermediate results.

Time Domain Interpretation of Effects of Heisenberg Transform

To get some intuition, it is beneficial to interpret the effect ofHeisenberg filtering in the time domain by exploring the structure of

(101 _(w)). Assuming w is factorizable, one can show that:

(Φ_(w))=w _(τ) *{W _(t)·

(Φ)},

where W_(t)=FT⁻¹ (w_(v))) and * stands for linear convolution. We seethat Heisenberg filtering amounts to a cascade of first applying awindow in time followed by a window in frequency, aka, convolution witha pulse (see FIG. 6). The main technical statement of this sectiondescribes the relation between the discrete and continuous ambiguityfunctions. The result will follow from the following generalproposition.

Proposition 5.1. Given a pair of Zak signals Φ₁Φ₂∈

and corresponding pair of Heisenberg filters w₁,w₂∈

(V), the following relation holds:

${_{\Phi_{w}} = {\sum\limits_{\lambda \in \Lambda}^{\;}\; {{_{\varphi}(\lambda)} \cdot P_{\lambda}}}},$

where w₂ ^(★)(v)=Ψ(β(v,v))w₂(−v) is the Heisenberg conjugate function.

In the case Φ₁ =Φ₂=Φwhere Φ=l(ϕ) and w₁=w₂=w the statement of theproposition describes the relation between the discrete ambiguityfunction of the sequence ϕ and the continuous ambiguity function of thewaveform Φ_(w). The result is summarized in the following theorem.

Theorem 5.2 (Main theorem of filter theory). Given a discrete Zak signalϕ∈

_(L) and a Heisenberg filter w∈

(V), the following relation holds:

${_{\Phi_{w}} = {\sum\limits_{\lambda \in \Lambda}{{_{\varphi}(\lambda)} \cdot P_{\lambda}}}},$

where P_(v)=w*_(σ)δ(v)*_(σ)w⁵⁶¹ for every v∈V.

In plain language, the theorem asserts that the ambiguity function ofthe waveform Φ_(w) is obtained from the ambiguity function of thesequence ϕ through shaping with a pulse P_(λ) (whose shape depends onthe particular value of λ). In a sense, the design of an optimal Radarwaveform involves two aspects. The first concerns the design of a finitesequence of a desired discrete ambiguity function and the secondconcerns the design of a Heisenberg filter w of a desired pulse shapeP_(λ) for various values of λ.

C6. Zak Theoretic Chirp Waveforms

In this section we describe a particular family of compressed Radarwaveforms based on discrete chirp sequences in the Zak domain. Thesewaveforms enjoy uniform temporal power profile and thumbtack likeambiguity function. The construction assumes the following set-up. Weassume N,M∈

are coprime odd integers. We let a∈(

/N)^(x) be an invertible element in the ring of integers modulo N. Wedenote by Ψ_(N):

/N→

the finite Fourier exponent Ψ_(N)(n)=Ψ(n/N).

We define the discrete Zak signal ch=ch_(a)∈

_(L) as:

${{ch}\left( {{n\; \Delta \; \tau},{m\; \Delta \; v}} \right)} = \left\{ {\begin{matrix}{\psi_{N}\left( {\frac{1}{2}{an}^{2}} \right)} & {m = {0\mspace{14mu} {mod}\mspace{14mu} M}} \\0 & {otherwise}\end{matrix},} \right.$

for every n,m␣1

. We refer to ch as the discrete Zak chirp of order N and slope a. Wenext explore the structure of the discrete ambiguity function

_(ch) . To that end, we introduce the sub-lattice Λ_(a)⊂Λ(see FIG. 11),given by:

Λ_(a)={(nΔτ,kMΔv):k=a·nmodN},

Theorem 6.1. The discrete ambiguity function

_(ch) is supported on the lattice Λ_(a) . Moreover:

_(ch)(nΔτ,kMΔv) =Ψ_(N)(½an ²)N,

for every (n,k) such that k=a·nmodN.

A direct consequence of Theorem 6.1 is that

_(ch) vanishes on the non-zero points of the intervalI_(r)=[−τ_(r)/2,τ_(r)2]×[−v_(r)/2,v_(r)/2], which we refer to as the“clean” region. Next, we fix a filter function w∈

(V) and define the continuous Zak chirp Ch=Ch_(a,2)∈

as:

Ch=w* _(σ) l(ch),

By the main theorem of filter theory (Theorem 5.2) we know that thecontinuous ambiguity function

_(ch) is related to the discrete ambiguity function

_(ch) through the equation:

$\begin{matrix}{_{Ch} = {\sum\limits_{\lambda \in \Lambda}{{_{ch}(\lambda)}P_{\lambda}}}} \\{{= {\sum\limits_{\lambda \in \Lambda_{a}}{{_{ch}(\lambda)}P_{\lambda}}}},}\end{matrix}\quad$

where P_(v)=w*_(σ)δ(v)*_(σ)w⁵⁶¹ for every v∈V. Assuming the pulses P_(λ)are well localized for every λ∈Λ_(a)

2I_(r), the continuous ambiguity function

_(ch) will have a thumbtack shape with a clean region around zerocoinciding with the interval I_(r) (see FIG. 12). In case the numbersN,M>>1, choosing he filter function w to be square root Nyquist withrespect to the lattice Λ ensures P_(λ) is well localized for every λ∈Λ

2I_(r).

Exemplary Methods Based on the Disclosed Technology

FIG. 13 is a flowchart of an example of a wireless communication method,and is described in the context of Section “C”. The method 1300includes, at step 1310, transforming an information signal to a discretesequence, where the discrete sequence is a Zak transformed version ofthe information signal. In some embodiments, the discrete sequence isquasi-periodic.

The method 1300 includes, at step 1320, generating a first ambiguityfunction corresponding to the discrete sequence. In some embodiments,the first ambiguity function is a discrete ambiguity function supportedon a discrete lattice.

The method 1300 includes, at step 1330, generating a second ambiguityfunction by pulse shaping the first ambiguity function. In someembodiments, the second ambiguity function is a continuous ambiguityfunction, and the pulse shaping is based on a pulse that is localized onthe discrete lattice.

The method 1300 includes, at step 1340, generating a waveformcorresponding to the second ambiguity function. In some embodiments, thewaveform includes a uniform temporal power profile.

The method 1300 includes, at step 1350, transmitting the waveform over awireless communication channel. While the processing performed in themethod 1300 is described as a number of steps, in general, it may bepossible to implement the input-to-output transformation withoutgenerating any intermediate signals explicitly. For example, thewaveform corresponding to the second ambiguity function may be directlygenerated from the information signal, without generating theintermediate discrete sequence or the first ambiguity function.

Accordingly, in another method for wireless communication, which isdescribed in the context of Section “C”, includes obtaining a waveformfrom an information signal, wherein the waveform corresponds to a secondambiguity function that is a pulse shaped version of a first ambiguityfunction, wherein the first ambiguity function corresponds to a discretesequence, and wherein the discrete sequence is a Zak transformed versionof the information signal, and transmitting the waveform over a wirelesschannel.

FIG. 14 is a flowchart of another example of a wireless communicationmethod, and is described in the context of Sections “A” and “B”. Themethod 1400 includes, at step 1410, transforming an information signalto a discrete lattice domain signal. In some embodiments, the discretelattice domain includes a Zak domain.

The method 1400 includes, at step 1420, shaping bandwidth and durationof the discrete lattice domain signal by a two-dimensional filteringprocedure to generate a filtered information signal. In someembodiments, the two-dimensional filtering procedure includes a twistedconvolution with a pulse. In other embodiments, the pulse is a separablefunction of each dimension of the two-dimensional filtering.

The method 1400 includes, at step 1430, generating, using a Zaktransform, a time domain signal from the filtered information signal. Insome embodiments, the time domain signal includes modulated informationsignal without an intervening cyclic prefix.

The method 1400 includes, at step 1440, transmitting the time domainsignal over a wireless communication channel. For example, a processormay implement the method 1400 and, at step 1440, may cause a transmittercircuit to transmit the generated waveform.

FIG. 15 shows an example of a wireless transceiver apparatus 1500. Theapparatus 1500 may be used to implement various techniques describedherein. The apparatus 1500 includes a processor 1502, a memory 1504 thatstores processor-executable instructions and data during computationsperformed by the processor. The apparatus 1500 includes reception and/ortransmission circuitry 1506, e.g., including radio frequency operationsfor receiving or transmitting signal and/or receiving data orinformation bits for transmission over a wireless network.

It will be appreciated that techniques for data modulation are disclosedin which information signal can be transmitted using multiple QAMsubcarriers without using a cyclic prefix. In some embodiments, amodulation technique, called OFDM-MultiCarrier (MC) may be used in whichQAM symbols are convolved with a periodic pulse function. In someembodiments, a Zak domain representation of a signal is used for shapingbandwidth and duration of a modulated information signal.

Exemplary Implementations of the Disclosed Technology

The disclosed and other embodiments, modules and the functionaloperations described in this document can be implemented in digitalelectronic circuitry, or in computer software, firmware, or hardware,including the structures disclosed in this document and their structuralequivalents, or in combinations of one or more of them. The disclosedand other embodiments can be implemented as one or more computer programproducts, i.e., one or more modules of computer program instructionsencoded on a computer readable medium for execution by, or to controlthe operation of, data processing apparatus. The computer readablemedium can be a machine-readable storage device, a machine-readablestorage substrate, a memory device, a composition of matter effecting amachine-readable propagated signal, or a combination of one or morethem. The term “data processing apparatus” encompasses all apparatus,devices, and machines for processing data, including by way of example aprogrammable processor, a computer, or multiple processors or computers.The apparatus can include, in addition to hardware, code that creates anexecution environment for the computer program in question, e.g., codethat constitutes processor firmware, a protocol stack, a databasemanagement system, an operating system, or a combination of one or moreof them. A propagated signal is an artificially generated signal, e.g.,a machine-generated electrical, optical, or electromagnetic signal, thatis generated to encode information for transmission to suitable receiverapparatus.

A computer program (also known as a program, software, softwareapplication, script, or code) can be written in any form of programminglanguage, including compiled or interpreted languages, and it can bedeployed in any form, including as a standalone program or as a module,component, subroutine, or other unit suitable for use in a computingenvironment. A computer program does not necessarily correspond to afile in a file system. A program can be stored in a portion of a filethat holds other programs or data (e.g., one or more scripts stored in amarkup language document), in a single file dedicated to the program inquestion, or in multiple coordinated files (e.g., files that store oneor more modules, sub programs, or portions of code). A computer programcan be deployed to be executed on one computer or on multiple computersthat are located at one site or distributed across multiple sites andinterconnected by a communication network.

The processes and logic flows described in this document can beperformed by one or more programmable processors executing one or morecomputer programs to perform functions by operating on input data andgenerating output. The processes and logic flows can also be performedby, and apparatus can also be implemented as, special purpose logiccircuitry, e.g., an FPGA (field programmable gate array) or an ASIC(application specific integrated circuit).

Processors suitable for the execution of a computer program include, byway of example, both general and special purpose microprocessors, andany one or more processors of any kind of digital computer. Generally, aprocessor will receive instructions and data from a read only memory ora random access memory or both. The essential elements of a computer area processor for performing instructions and one or more memory devicesfor storing instructions and data. Generally, a computer will alsoinclude, or be operatively coupled to receive data from or transfer datato, or both, one or more mass storage devices for storing data, e.g.,magnetic, magneto optical disks, or optical disks. However, a computerneed not have such devices. Computer readable media suitable for storingcomputer program instructions and data include all forms of non-volatilememory, media and memory devices, including by way of examplesemiconductor memory devices, e.g., EPROM, EEPROM, and flash memorydevices; magnetic disks, e.g., internal hard disks or removable disks;magneto optical disks; and CD ROM and DVD-ROM disks. The processor andthe memory can be supplemented by, or incorporated in, special purposelogic circuitry.

While this patent document contains many specifics, these should not beconstrued as limitations on the scope of an invention that is claimed orof what may be claimed, but rather as descriptions of features specificto particular embodiments. Certain features that are described in thisdocument in the context of separate embodiments can also be implementedin combination in a single embodiment. Conversely, various features thatare described in the context of a single embodiment can also beimplemented in multiple embodiments separately or in any suitablesub-combination. Moreover, although features may be described above asacting in certain combinations and even initially claimed as such, oneor more features from a claimed combination can in some cases be excisedfrom the combination, and the claimed combination may be directed to asub-combination or a variation of a sub-combination. Similarly, whileoperations are depicted in the drawings in a particular order, thisshould not be understood as requiring that such operations be performedin the particular order shown or in sequential order, or that allillustrated operations be performed, to achieve desirable results.

Only a few examples and implementations are disclosed. Variations,modifications, and enhancements to the described examples andimplementations and other implementations can be made based on what isdisclosed.

What is claimed is:
 1. A method for wireless communication, comprising:transforming an information signal to a discrete sequence, wherein thediscrete sequence is a Zak transformed version of the informationsignal; generating a first ambiguity function corresponding to thediscrete sequence; generating a second ambiguity function by pulseshaping the first ambiguity function; generating a waveformcorresponding to the second ambiguity function; and generating thewaveform for transmission over a wireless communication channel.
 2. Themethod of claim 1, wherein the waveform comprises a uniform temporalpower profile.
 3. The method of claim 1, wherein the first ambiguityfunction is a discrete ambiguity function supported on a discretelattice.
 4. The method of claim 3, wherein the second ambiguity functionis a continuous ambiguity function, and wherein the pulse shaping isbased on a pulse that is localized on the discrete lattice.
 5. A methodfor wireless communication, comprising: obtaining a waveform from aninformation signal, wherein the waveform corresponds to a secondambiguity function that is a pulse shaped version of a first ambiguityfunction, wherein the first ambiguity function corresponds to a discretesequence, and wherein the discrete sequence is a Zak transformed versionof the information signal; and transmitting the waveform over a wirelesschannel.
 6. The method of claim 5, wherein the waveform comprises auniform temporal power profile.
 7. The method of claim 5, wherein thefirst ambiguity function is a discrete ambiguity function supported on adiscrete lattice.
 8. The method of claim 7, wherein the second ambiguityfunction is a continuous ambiguity function, and wherein the pulseshaping is based on a pulse that is localized on the discrete lattice.9. The method of claim 5, wherein the discrete sequence isquasi-periodic.
 10. A method for wireless communication, comprising:transforming an information signal to a discrete lattice domain signal;shaping bandwidth and duration of the discrete lattice domain signal bya two-dimensional filtering procedure to generate a filtered informationsignal; generating, using a Zak transform, a time domain signal from thefiltered information signal; and transmitting the time domain signalover a wireless communication channel.
 11. The method of claim 10,wherein the discrete lattice domain includes a Zak domain.
 12. Themethod of claim 10, wherein the two-dimensional filtering procedureincludes a twisted convolution with a pulse.
 13. The method of claim 11,wherein the pulse is a separable function of each dimension of thetwo-dimensional filtering.
 14. The method of claim 10, wherein the timedomain signal comprises modulated information signal without anintervening cyclic prefix.